Chapter 9 Infinite Series
|
|
- Scott Howard Atkinson
- 5 years ago
- Views:
Transcription
1 Sctio Cotiud d + d + C Ar lim b lim b b b + b b lim + b b lim + b b 6. () d (b) lim b b d (c) Not tht d c b foud by prts: d ( ) ( ) d + C. b Ar b b lim d lim b b b b lim ( b + ). b dy 7. () π dy π ( y+ ) (b) lim π b dy b ( y+ ) (c) Volum lim π b b dy ( y+ ) lim π ( y+ ) b lim π + + π b b b 8. Not tht d c b foud by prts: d d ( ) ( ) + C. So k k d lim d lim k k lim + k k k k k By L Hopitl s rul, lim lim. k k k k k Thrfor, d lim + k k k + Th itgrl covrgs to. Chptr 9 Ifiit Sris Sctio 9. Powr Sris (pp. 7 8) Eplortio Fidig Powr Sris for Othr Fuctios. + + ( ) ( ) ( ).. ( ) + ( ) ( ) + ( ) ( ). This gomtric sris covrgs for < <, which is quivlt to < <. Th itrvl of covrgc is (, ).. ( ) + ( ) ( ) + ( ). This gomtric sris covrgs for < <, which is quivlt to < <. Th itrvl of covrgc is (, ). Eplortio Fidig Powr Sris for t ( ).. t dt + t ( ( ) t t t t ) dt 7 + t t t t ( ) t ( ) Th grphs of th first four prtil sums ppr to b covrgig o th itrvl (, ).
2 78 Sctio 9.. Wh, th sris bcoms ( + ) +. This sris dos ppr to covrg. Th trms r gttig smllr, d bcus thy ltrt i sig thy cus th prtil sums to oscillt bov d blow limit. Th two clcultor sttmts show blow will cus th succssiv prtil sums to ppr o th clcultor ch tim th ENTER butto is pushd. Th prtil sums will ppr to b pprochig limit of π / (which is t ( )), lthough vry slowly. 7. Th t thr prtil sums show tht th covrgc tds outsid th itrvl (, ) i both dirctios, so (, ) ws pprtly udrstimt. Your swr i #6 might hv b bttr, but ulss you gussd ll rl umbrs you still udrstimtd! (S Empl i Sctio 9..) Eplortio A Sris with Curious Proprty. f ( ) !!!. f ( ) Sic this fuctio is its ow drivtiv d tks o th vlu t, w suspct tht it must b. dy. If y f( ), th y d y wh. d. Th diffrtil qutio is sprbl. dy d y dy d y l y + C k y K K K y. 6. Th first thr prtil sums r show i th grph blow. It ir risky to drw y coclusios bout th itrvl of covrgc from just thr prtil sums, but so fr th covrgc to th grph of y oly looks good o (, ). Your swr might diffr. Quick Rviw 9.. u + u + u + u + 6 u + 8. u u u u u ( ) ( ) ( ) ( ) ( ). () Sic 6 8, th commo rtio is (b) ( ) 9, 66 (c) ( ). () Sic, th commo rtio is. 8 (b) (c) 8 8(. )
3 Sctio () W grph th poits, for,,,. (Not tht thr is poit t (, ) tht dos ot show i th grph.) 9. () W grph th poits,,,,. for (b) lim [, ] by [.,.] lim 6. () W grph th poits, +,,,. for (b) lim [,.] by [, ] lim. () W grph th poits, l( + ) for,,,. (b) lim [,.] by [, ] lim + 7. () W grph th poits (,( ) )for,,,. (b) lim [,.] by [, ] dos ot ist bcus th vlus of oscillt btw d. 8. () W grph th poits,,,,. + for [,.] by [, ] (b) lim lim + [,.] by [, ] (b) lim lim l( + ) lim ( + ) Sctio 9. Erciss. () Lt u rprst th vlu of i th th-trm, strtig with. Th,,, u u u 9 d, so u 6 u, u, u 9, d u 6. W my writ u, or. (b) Lt u rprst th vlu of i th th-trm, strtig with. Th,,, d u u u 9, so u, u, u 9, d u 6. W u 6 my writ u ( + ), ( + ) or. (c) If, th sris is ( ) ( ) ( ) ( 6 ), which is th sm s th dsird sris. Thus lt.
4 8 Sctio 9.. () Not tht,,, d so o. Thus 9. (b) Not tht,,, d so o. Thus ( ). (c) Not tht,.,., d so o. Thus (. ).. Diffrt, sic th trms of ltrt btw positiv d gtiv, whil th trms of ll gtiv.. Th sm, sic both sris c b rprstd s Th sm, sic both sris c b rprstd s Diffrt, sic + but 8 ( ) Divrgs bcus th trms do ot pproch zro. 8. Divrgs bcus th trms ltrt btw d. 9. Covrgs bcus th trms pproch zro.. Covrgs bcus th trms pproch zro.. Covrgs;. Divrgs, bcus th trms do ot pproch zro.. Covrgs;. Divrgs, bcus th commo rtio is d th trms do ot pproch zro.. Divrgs, bcus th trms ltrt btw d d do ot pproch zro. r 7. Covrgs; si π + π ( ) ( + )( ) 8. Divrgs, bcus th trms do ot pproch zro. 9. Covrgs; sic. 86 <, π π π π π. Covrgs; 6 +. Sic ( ) , th sris covrgs wh < d th itrvl of covrgc is,. Sic th sum of th sris is fuctio f( ), < <, th sris rprsts th.. Sic ( ) ( + ) [ ( + )], th sris covrgs wh ( + ) < d th itrvl of covrgc is (, ). Sic th sum of th sris is, [ ( + )] + th sris rprsts th fuctio f( ), < <. +. Sic ( ) covrgs wh, th sris < d th itrvl of covrgc is (, ). Sic th sum of th sris is, ( ) th sris rprsts th fuctio f( ), < <. 6. Covrgs; (. ) (. )
5 Sctio For, th sris covrgs wh < d th itrvl of covrgc is (, ). Sic th sum of 6 th sris is, th sris rprsts th ( ) 6 fuctio f( ), < <.. Sic si (si ), th sris covrgs wh si <. Thus, th sris covrgs for ll vlus of cpt odd itgr multipls of π, tht is, ( k+ ) π for itgrs k. Sic th sum of th sris is si, th sris rprsts th fuctio f ( ) si, ( k + ) π. 6. Sic t (t ), th sris covrgs wh t <. Thus, th sris covrgs for π π + kπ < < + k π, whr k is itgr. Sic th sum of th sris is, th sris rprsts th fuctio t f( ) k k t, π π + π < < + π. d d 7. f ( ) ( d ( ) d ) f ( ) ( ) f ( ), < < ( ) d d 8. f ( ) ( ( ) d ( )( + ) d + + ( + ) ( + ) + ( ) ( + ) ) f ( ) + ( + ) ( + ) ( + ) + ( ) ( + ) f ( ) ( ) ( + ), < < ( + ) d 9. f ( ) d ( ) d ( d + ) ( ) ( ) ( ) f ( ) + ( ) ( ) ( ) 8 ( ) f ( ) ( ), < < ( ) d d. f d d + ( ) f ( ) ( ) f ( ) ( ) t dt ( t t t t ) dt l( ) l( ), < < / + + dt. ( t + ) ( ( t+ ) + ( t+ ) ( t+ ) + + ( ) ( t+ ) ) dt l( + ) ( + ) + ( + ) ( + ) ( ) ( + ) + ( ) + l( + ) ( + ) +
6 8 Sctio 9.. dt t ( t ) + ( t) ( t) ( t ) dt l ( ) + ( ) ( ) ( ) + + ( ) l, < < +. 6 dt ( t ) t + + t t ( t ) + 6 dt ( ) l t l( ) +. () Sic th trms r ll positiv d do ot pproch zro, th prtil sums td towrd ifiity. (b) Th prtil sums r ltrtly d. (c) Th prtil sums ltrt btw positiv d gtiv whil thir mgitud icrss towrd ifiity. 6. Sic π π π, this is gomtric sris with 6 π π commo rtio r.,which is grtr th o. π 7., < O possibl swr: For y rl umbr, us To gt, 8 6 us Assumig th sris bgis t ; () r, r < r r r r Sris: (b) r, r < r Sris: r r r. Lt d r, givig.. +. (. ) +. (. ) +. (. ). (. ) Lt d r, givig.. +. (. ) +. (. ) +. (. ). (. )
7 Sctio (.) + 7.(.) + 7.(.) + 7.(.) d.. d [ ] d (. ) d. d 9. d (. ) + 6. (. ) + 6. (. ) + 6. (. ) (. ) +. (. ) +. (. ) (. ) +. (. ). +. (. ) ,,, ( ) (. 87) Totl distc + 6 [(.) + 6 (.) + 6 (.) ] + 6. (. ). + i. 6 + i 6 6m 9. Totl tim 6 (. ) 6 (. ) 6 (. ) (. ) [. (. ) ].. + (. 6) i sc. Th r of ch squr is hlf of th r of th prcdig squr, so th totl of ll th rs is 8m.. Totl r i π i π π π i π
8 8 Sctio 9.. () S rs ( + r + r + r + r + r ) ( r + r + r + r + r + r ) r (b) Just fctor d divid by r: S rs r S( r) r r S r. Usig th ottio S + r + r + r + r r formul from Ercis is S. r If r <, th lim r d so r r lim S lim. r r, th If r > or r, th r hs o fiit limit s, so th prssio r hs o fiit limit d r r divrgs. If r, th th th prtil sum is, which gos to ±.. Comprig with, th ldig trm is + r d th commo rtio is r. Sris: ( ) Itrvl: Th sris covrgs wh itrvl of covrgc is,. <, so th. Comprig with, th first trm is r d th commortiois r. Sris: Itrvl: Th sris covrgs wh <, so th itrvl of covrgc is,. 6. Comprig with r commo rtio is r. 6 Sris: + + +, th first trm is d th Itrvl: Th sris covrgs wh <, so th itrvl of covrgc is (, ). 7. Comprig with, th first trm is + ( ) r d th commo rtio is r ( ). Sris: ( ) + ( ) + ( ) ( ) Itrvl: Th sris covrgs wh <, so th itrvl of covrgc is (, ). 8. Comprig + ( ) with r, th first trm is d th commo rtio is r ( ). Sris: ( ) + ( ) + ( ) ( ) Itrvl: Th sris covrgs wh <, so th itrvl of covrgc is (, ). 9. Rwritig s d comprig with ( ) th first trm is d th commo rtio is r Sris: + ( ) + ( ) + ( ) r,. Itrvl: Th sris covrgs wh <, so th itrvl of covrgc is (, ). Altrt solutio: Rwritig s d comprig with r, th first is d th commo rtio is r. Sris: Itrvl: Th sris covrgs wh of covrgc is (, ). b b b b ( ) <, so th itrvl 9 b b 99 b 9 8 b b l l l 6. () Wh t, S. (b) S covrgs wh t + t <, or t < + t. For t <, this iqulity is quivlt to t < ( + t), which is lwys fls. For t <, th iqulity is quivlt to t < + t, which is tru wh t >.
9 Sctio Cotiud (b) For t, th iqulity is quivlt to t < + t, which is lwys tru. Thus, S covrgs for ll t >. (c) For t >, w hv t + t S t t t + + t t +, so S > ( ) + t wh t > () Th sris covrgs to S ms tht lim S S, whr S k k (b) S ( + ) + S + k is th th prtil sum of th sris. (c) lim S lim S + lim so S S+ lim or lim 6. Sic ( ) + ( ) ( ) + ( ) ( ), w my writ l t dt ( ) ( ) + ( ) ( + ) ( ) 6. To dtrmi our strtig poit, w ot tht f ( ) d ( ) d ( ) + C. Usig th rsult from Empl, w hv: ( ) d d ( ) ( ) d d ( ) ( ) Thus, f( ) ( ) 6. Rplcig by +, this my b writt s f( ) ( + )( + ). Itrvl: Th sris covrgs wh <, so th itrvl of covrgc is (, ). 6. () No, bcus if you diffrtit it gi, you would hv th origil sris for f, but by Thorm, tht would hv to covrg for < <, which cotrdicts th ssumptio tht th origil sris covrgs oly for < <. (b) No, bcus if you itgrt it gi, you would hv th origil sris for f, but by Thorm, tht would hv to covrg for < <, which cotrdicts th ssumptio tht th origil sris covrgs oly for < <. 66. Fls. It divrgs bcus it is gomtric sris with rtio. tht is grtr th. 67. Fls. It covrgs bcus it is gomtric sris with rtio tht is lss th. 68. C. / / 69. A., < < ( ) 7. E. 7. D. t dt t l l 7. () Comprig f() t + t d th commo rtio is r First four trms: t + t t Grl trm: ( ) ( t ) with, th first trm is r 6 t. (b) Not tht G(), so th costt trm of th powr sris for G() will b. Itgrt th trms for f() to obti th trms for G(). 7 First four trms: + 7 Grl trm: ( ) + + (c) Th sris i prt () covrgs wh t <, so th itrvl of covrgc is (, ). (d) Th two umbrs r ±, which rsult i th covrgt sris G() + + ( ) d G( ) ( ), 7 + rspctivly. 7. Lt L lim. Th by dfiitio of covrgc, for thr corrspods N such tht for ll m d, m, > N m L< d L<. Now, m m L+ L m L + L < + whvr m > N d > N.
10 86 Sctio Giv >, by dfiitio of covrgc thr corrspods N such tht for ll < N, L < d L <. (Thr is o such umbr for ch sris, d w my lt N b th lrgr of th two umbrs.) Now L L L + L L + L < +. L L < sys tht th diffrc btw two fid vlus is smllr th y positiv umbr. Th oly ogtiv umbr smllr th vry positiv umbr is, so L L or L L. 7. Cosidr th two subsqucs d whr k ( ) i ( ), lim L, lim L, d L L. Giv > k ( ) i ( ) thr corrspods N such tht for k() > N, k ( ) L <, d N such tht for i ( ) N, > i ( ) L <. Assum covrgs. Lt N m { N, N}. Th for > N, w hv tht L < d L < for ifiitly my. This implis tht lim L d lim L whr L L. Sic th limit of squc is uiqu (by Ercis ), dos ot covrg d hc divrgs () lim + (b) Th li y is horizotl symptot of th grph of + th fuctio f( ), which ms lim f( ). + Bcus f( ) for ll positiv itgrs, it follows tht lim must lso b. Sctio 9. Tylor Sris (pp. 8 9) Eplortio Dsigig Polyomil to Spcifictios. Sic P(), w kow tht th costt cofficit is. Sic P (), w kow tht th cofficit of is. Sic P (), w kow tht th cofficit of is. (Th i th domitor is dd to ccl th fctor of tht rsults from diffrtitig.) Similrly, w fid th cofficits of d to b d. 6 Thus, P ( ) Eplortio A Powr Sris for th Cosi. cos( ) cos ( ) si( ) cos ( ) cos( ) ( ) cos ( ) si( ) tc. Th pttr,,, will rpt forvr. Thrfor, 6 P 6 ( ) +! 6!, d th Tylor sris is ( ).! 6! ( )!. A clvr shortcut is simply to diffrtit th prviouslydiscovrd sris for si trm-by-trm! Eplortio Approimtig si trms. Quick Rviw 9.. f( ) f ( ) f ( ) f ( ) 8 ( ) f ( ). f( ) f ( ) ( ) f ( ) ( ) f ( ) 6( ) f ( ) ( )!( ) + ( ) ( ). f( ) f ( ) l f ( ) (l ) f ( ) (l ) ( f ) ( ) (l ). f( ) l( ) f ( ) f ( ) f ( ) ( ) f ( ) 6 ( f ) ( ) ( ) ( )! for
11 Sctio f( ) f ( ) f ( ) ( ) f ( ) ( )( ) ( k)! f ( ) ( k)! k ( )! f ( )!! 6. dy d 7. dy d 8. dy d 9. dy d. dy d d d!! ( )! d d d ( ) d d d d d ( ) ( )!! ( + ) ( )! + + ( + ) ( ) ( )! ( + )! ( + ) ( )! ( ) ( )! ( + ) ( )! ( ) ( )! ( ) ( ) ( ) ( )!! ( )! Sctio 9. Erciss. P( ) + P ( ) + P ( ) / ( + ) P ( ) / ( + ) ( ) ( ) P ( ) 7/ ( + ) P ( ) P( ) P ( ) P ( ) P ( ) 8 8 ( ) P ( ) 6 6 P ( ) P( ) + P ( ) ( + ) P ( ) ( + ) 6 P ( ) ( + ) 8 ( ) P ( ) ( + ) ( ) P ( ) 6 ( + ) 8 P ( ) + + +, 6 ( ). P( ) P ( ) P ( ) P ( ) ( ) P ( ) ( ) P ( ) P ( ) + + +, ( )!!!!!. Substitut for i th Mcluri sris for si show t th d of Sctio ( ) ( ) ( ) si + + ( )!! ( + )! ( ) ( ) + ( + )! This sris covrgs for ll rl. 6. Substitut for i th Mcluri sris for l(+) show t th d of Sctio 9.. ( ) ( ) ( ) l( ) ( ) + + ( ) This sris covrgs wh <, so th itrvl of covrgc is [, ). 7. Substitut for i th Mcluri sris for t show t th d of Sctio ( ) ( ) ( ) t + + ( ) ( + ) + This sris covrgs wh, so th itrvl of covrgc is [, ].
12 88 Sctio ( )!! !! This sris covrgs for ll rl. 9. cos ( + ) (cos ) (cos ) (si ) (si ) (cos ) + + ( )!! ( )! + (si ) + + ( )!! ( + )! (cos ) (si ) (cos ) (cos ) (si ) + +!!! (si )! W d to writ prssio for th cofficit of k. If k is v, th cofficit is ( ) (cos ) whr k. ( )! Thus th cofficit is ( ) k / (cos ), which is th sm s k! it[( k+ ) / ] ( ) (cos ). If k is odd, th cofficit is k! + ( ) (si ) whr + k. Thus th cofficit is ( + )! ( k+ ) / ( ) (si ), which is th sm s ( + )! it[( k+ )/ ] ( ) (cos ). Hc th grl trm is k! ( ) A B +, whr A it!, d B si if is odd d B cos if is v. Aothr wy to hdl th grl trm is to obsrv tht si cos + π, cos cos ( + π ), d so o, so th grl trm is! cos + π. Th sris covrgs for ll rl.. cos + + ( )!! ( )! 6 + ( + + ) ( )! Th sris covrgs for ll rl.. Fctor out d substitut for i th Mcluri sris for show t th d of Sctio 9.. [ + + ( ) + ( ) ] Th sris covrgs for <, so th itrvl of covrgc is (, ).. Substitut for i th Mcluri sris for show t th d of Sctio ( ) ( + + ) ( )!! + + ( )! Th sris covrgs for ll rl.. P( ) ( ) ( ) P ( ) ( ) + ( ) ( ) +. P( ) ( ) P ( )! ( ) ( + + ) ( ) ( ) + +!!!. () Sic f is cubic polyomil, it is its ow Tylor polyomil of ordr. P ( ) + or + (b) f() + f () f () f () 6 6, so! f f () () 6 6, so! P ( ) + ( ) + ( ) + ( ) 6. () Sic f is cubic polyomil, it is its ow Tylor polyomil of ordr. P ( ) or (b) f() f () f () f () +, so 7! f () f (), so! P ( ) + ( ) + 7( ) + ( )
13 Sctio () Sic f( ) f ( ) f ( ) f ( ), th Tylor polyomil of ordr is P ( ). () (b) f f () f () f (), so 6! f f () (), so! P ( ) + ( ) + 6( ) + ( ) 8. f ( ) f ( ) f ( ) f ( ), so! 8 f ( ) f ( ) 6, so 8! 6 P ( ) P ( ) ( ) P ( ) + 8 ( ) ( ) P ( ) π 9. f si π f π cos π f π si π π f cos π π f, so! π f, so! P ( ) P ( ) + π P ( ) + π π P () + π π π π. f cos π f π si π f π π f cos, so π! f π f π si, so π! P ( ) P ( ) π π P ( ) π P ( ) π π + π. f( ) f ( ) f ( ) f ( ), so! 6 f ( ) f ( ), so 8 6! P ( ) P ( ) + ( ) + ( ) P 6 ( ) ( ) P ( ) () P ( ) + + +!! + + f(. ) P (. ) 88. (b) Sic th Tylor sris of f ( ) c b obtid by diffrtitig th trms of th Tylor sris of f( ), th scod ordr Tylor polyomil of f ( ) is giv by 8+. Evlutig t., f (.).. () P ( ) + ( )( ) +! ( ) +! ( ) ( ) + ( ) + ( ) f(.) P (.). 86
14 9 Sctio 9.. Cotiud (b) Sic th Tylor sris of f ( ) c b obtid by diffrtitig th trms of th Tylor sris of f( ),th scod ordr Tylor polyomil of f ( )is giv by + ( ) + ( ). Evlutig t., f (.) 6.. () Sic f ( ) f!, ( ).! Sic f ( ) ( ) ( ) f!!, ( )!.! (b) Multiply ch trm of f( ) by. + g ( ) !!! ( + )! (c) g ( ). () Substitut for i th Mcluri sris for show t th d of Sctio 9. (b) g ( ) + + +! i! !!! + + +!!! + + +!!! This c lso b writt s !! ( + )! d ( )( ) ( )( ) (c) g ( ) d + g + () From th sris, d g ( ) d!!! ( + )! + + +!!! ( + )! ( + )! Thrfor, g () ( + )!, which ms. ( + )! 6. () Fctor out d substitut t for i th Mcluri sris for t th d of Sctio 9.. f() t t t + t + ( t ) + ( t ) + ( t ) 6 + t + t + t + t (b) Sic G( ), th costt trm is zro d w my fid G( )by itgrtig th trms of th sris for f( ). 7 + G( ) () f ( ) ( + ) f ( ) ( + ) f ( ) ( + ) ( ), so f! ( ) f ( ) ( ), so f 8! 6 P ( ) (b) Sic g ( ) f ( ), th first four tms r (c) Sic h( ), th costt trm is. Th t thr trms r obtid by itgrtig th first thr trms of th swr to prt (b). Th first four tms of th sris for h ( ) r () i 9 i 9
15 Sctio Cotiud () Sic ch trm is obtid by multiplyig th prvious trm by,! ! ( ) (b) Sic th sris c b writt s, it! rprsts th fuctio f( ). (c) f () 9. First, ot tht cos Usig cos ( ) ( )!, tr th followig two-stp commds o your hom scr d cotiu to hit ENTER. Th sum corrspodig to N is bout.68 (ot withi. of th ct vlu), d th sum corrspodig to N 6 is bout.666, which is withi. of th ct vlu. Sic w bg with N, it tks totl of 7 trms (or, up to d icludig th d dgr trm).. O possibl swr: Bcus th d bhvior of polyomil must b uboudd d si is ot uboudd. Aothr: Bcus si hs ifiit umbr of locl trm, but polyomil c hv oly fiit umbr.. () si is odd d cos is v () si d cos. Rplc by i sris for si. Thrfor, w hv ( ) 8 so.!!. Sic d l, which is d is.! t, th cofficit. Th liriztio of f t is th first ordr Tylor polyomil grtd by f t. d. () Sic f ( ) d + ( + )( ) ( )( ) ( + ) ( + ), w hv f( ), f ( ), f( ) d f ( ), so th liriztios r L( ) d L ( ) ( ) +, rspctivly. [, ] by [, ] (b) f ( ) must b bcus of th iflctio poit, so th scod dgr trm i th Tylor sris of f t is zro. 6. Th sris rprsts t. Wh, it covrgs to t π. Wh, it covrgs to π t ( ). 7. Tru. Th costt trm is f ( ). 8. Fls. It is bcus th cofficit of is f ( ).! 9. E. f( ) + + +! f ( ) +. A.!. C.. A () f ( ) (si ) ( )!! ( + )! + + ( )!! ( + )! (b) Bcus f is udfid t. (c) k. Not tht th Mcluri sris for is If w diffrtit this sris d multiply by, w obti th dsird Mcluri sris Thrfor, th dsird fuctio is f d ( ) d ( ) ( ).
16 9 Sctio 9.. () f( ) ( + ) m m f ( ) m( + ) m f ( ) m( m )( + ) f ( ) m( m)( m )( + ) m (b) Diffrtitig f() k tims givs ( k f ) mk ( ) m( m)( m) ( m k+ )( + ). Substitutig or, w hv f ( k ) ( ) m ( m )( m ) ( m k+ ). (c) Th cofficit is ( k f ) ( ) mm ( )( m) ( m k+ ) k! k! (d) f( ), f ( ) m, d w r do by prt (c). 6. Bcus f( ) ( + ) m is polyomil of dgr m. Altrtly, obsrv tht f ( k ) ( ) for k m+. Sctio 9. Tylor s Thorm (pp. 9 ) Eplortio Your Tur. W d to cosidr wht hpps to R ( ) s. By ( +) f () c Tylor s Thorm, R ( ) ( )! ( + ), whr ( + ) () f c is th ( +stdrivtiv ) of cos vlutd t som c btw d. As with si, w c sy tht ( + ) () f c lis btw d iclusiv. Thrfor, o mttr wht is, w hv ( +) f () c R ( ) ( )! ( ) + ( + )! ( + )!. Th fctoril growth i th domitor, s otd i Empl, vtully outstrips th powr growth i th umrtor, d w hv for ll. This ms tht R ( + )! ( ) for ll, which complts th proof. Eplortio Eulr s Formul i ( i) ( i) ( i). + i+ + +!!! + i i + + i + () i!!!!!. If w isolt th trms i th sris tht hv i s fctor, w gt: i i i i () i!!!!! ( ) + i!! 6! ( )!! ( )!! 7 ( + )! cos + isi. (W r ssumig hr tht w c rrrg th trms of covrgt sris without ffctig th sum. It hpps to b tru i this cs, but w will s i Sctio 9. tht it is ot lwys tru.) iπ. cosπ + isiπ + Thus, iπ + Quick Rviw 9.. Sic f( ) cos( ) o [ π, π] d f( ), M.. Sic f() is icrsig d positiv o [, ], M f( ) 7.. Sic f() is icrsig d positiv o [, ], M f( ).. Sic th miimum vlu of f( ) is f( ) d th mimum vlu of f( ) is f( ), M.. O [, ], th miimum vlu of f( ) is f( ) 7 d th mimum vlu of f( ) is f( ). O (, ], f is icrsig d positiv, so th mimum vlu of f is f (). Thus f( ) 7 o [, ] d M Ys, sic ch prssio for th drivtiv giv by th Quotit Rul will b rtiol fuctio whos domitor is powr of No, sic th fuctio f( ) hs corr t. 8. Ys, sic th drivtivs of ll ordrs for si d cos r dfid for ll vlus of. 9. Ys, sic th fuctio f( ) hs drivtivs of th ( form f ) ( ) ( for odd vlus of d f ) ( ) for v vlus of, d both of ths prssios r dfid for ll vlus of.. No, sic f( ), w hv f ( ) d f ( ), so f ( ) is udfid.
17 Sctio 9. 9 Sctio 9. Erciss. f( ) f ( ) f ( ) f ( ), so! f ( ) f ( ) 8 8, so! ( ) ( ) ( ) f ( ) 6 f 6, so! P ( ) + + f(. ) P (. ). 67 π. f ( ) cos π π f ( ) si π π π ( ) π f ( ) cos, so f! 8 π π ( ) f ( ) si, so f 8! ( ) ( ) π π π ( ) f ( ) cos, so f π 6 6! 8 π π P ( ) f(. ) P (. ). 9. f( ) si( ) si f ( ) cos f ( ) f ( ) si, so! f ( ) f ( ) cos, so! 6 ( ) ( ) f ( ) f ( ) si, so! P ( ) f(. ) P (. ). 99. Substitutig for i th Mcluri sris giv for l( + ) t th d of Sctio 9., w hv ( ) ( ) ( ) l( + ) + + ( ) ( ) Thrfor, P ( ) d f(. ) P (. ). 9.. f( ) ( ) f ( ) ( ) f ( ) 6( ) 6, so f ( )! f ( ) f ( ) ( ), so! ( ) ( ) 6 f ( ) f ( ) ( ), so! P ( ) f(. ) P (. ) si +! ( ) +!! ( )! +! ( )! 7! 9! ( + )! Not: By rplcig with +, th grl trm c b + writt s ( ) ( + )! !! !! 8. cos + cos ( ) ( ) ( ) ( ) ( )!! ( )! ( ) i! i! i ( )! ( ) + + ( )! 9. si cos( ) + ( ) ( ) ( )!! 6! 6 6 ( ) + ( ) ( )! ( ) i! i! i 6! i ( )! ( ) + ( )! Not: By rplcig with +, th grl trm c b + + writt s ( ) ( + )!.
18 9 Sctio 9.. [ + + ( ) + ( ) ] P 7 ( ). P ( ). ( ). 7 ( + ) t + (). f t t Mr t M t [, ] M [, ] M t t M + + ( t 6. f ) () t sit t+ Mr! t M sit t+! [, ] M [, ] M si +! t sit t + M! ( 7. f ) () t si t Mr + + M si t [, ] M [, ] M si si t M ( 8. f ) () t cos t Mr + + M cos t [, ] M [, ] M cos cos t M 9. Lt f( ) si. Th P( ) P( ), so w us 6 th Rmidr Estimtio Thorm with. Sic ( f ) ( ) cos for ll, w my us M r, givig R ( )!, so w my ssur tht R ( ) by rquirig, or! Thus, th bsolut rror is o grtr th wh 6. < < 6. (pproimtly). Altrt mthod: Usig grphig tchiqus, si wh 7. < < Lt f( ) cos. Th P( ) P( ), so w my us th Rmidr Estimtio Thorm with. Sic ( f ) ( ) cos for ll, w my us M r, givig R ( )!. For <., th bsolut rror is lss th (. ). 6 (pproimtly).! Altrt mthod: Usig grphig tchiqus, w fid tht wh <., rror cos. < cos.. 8. Th qutity grphs of y cos d y. tds to b too smll, s show by th. Lt f( ) si. Th P( ) P( ), so w my us th Rmidr Estimtio Thorm with. Sic f ( ) cos for ll, w my us M r, givig R ( )!. Thus, for <, th mimum possibl rror is bout ( ) 67..! Altrt mthod: Usig grphig tchiqus, w fid tht wh <, rror si si. 67.
19 Sctio Cotiud Th iqulity < si is tru for <, s w my s by grphig y si.. Lt f( ) +. Th P ( ) +, so w my us th Rmidr Estimtio Thorm with. Sic f ( ) ( + ), which is lss th.8 for <., w my us M. 8 d r, givig. 8 R ( ). Thus, for <. th mimum! Possibl bsolut rror is bout Altrt mthod:. 8(. )! 7.. Usig grphig tchiqus, w fid tht wh <., rror Not tht + + is th scod ordr Tylor polyomil for f( ) t, so w my us th Rmidr Estimtio Thorm with. Sic f ( ), which is lss th. wh <. d r, givig. R ( ). Thus, for <.,th mimum possibl!. rror is bout (. ). 8.!. Not tht d!! + + ( ) Thus th trms with!! v will ccl for sih ( ), d th trms with odd will ccl for cosh ( ). + sih !! ( + )! cosh !! ( )!. All of th drivtivs of cosh r ithr cosh or sih. For y rl, cosh d sih r both boudd by. So for y rl, lt M d r i th Rmidr + Estimtio Thorm. This givs R ( ) But for ( + )! + y fid vlu of, lim. It follows tht th ( + )! sris covrgs to cosh for ll rl vlus of. 6. For, Tylor s Thorm with Rmidr sys tht if f hs drivtivs of ll ordrs i op itrvl I cotiig, th for ch i I, f( ) f( ) + R( ), whr R ( ) f ( c)( ), so f( ) f( ) + f ( c)( ) w for som c btw d. Lttig b this qutio is f( b) f( ) + f ( c)( b ), which is quivlt to f( b) f( ) f () c for som c btw d b. Thus, for b th clss of fuctios tht hv drivtivs of ll ordrs i op itrvl cotiig d b, th M Vlu Thorm c b cosidrd spcil cs of Tylor s Thorm. 7. f( ) l(cos ) l f ( ) ( si ) t cos f ( ) sc f () so! () L () (b) P ( ) (c) Th grphs of th lir d qudrtic pproimtios fit th grph of th fuctio r. [, ] by [, ] si 8. f( ) si f ( ) cos si f ( ) si ( )( si ) + (cos )( cos, f ( ) so! () L ( ) + (b) P ( ) + +
20 96 Sctio Cotiud (c) Th grphs of th lir d qudrtic pproimtios fit th grph of th fuctio r. () L () (b) P ( ) (c) Th grphs of th lir d qudrtic pproimtios fit th grph of th fuctio r. [, ] by [,] 9. f( ) ( ) f ( ) ( ) ( ) ( ) f + ( ) ( ) ( ) ( ) ( ), so f ( )! () L( ) (b) P ( ) + (c) Th grphs of th lir d qudrtic pproimtios fit th grph of th fuctio r. [, ] by [, ]. f( ) sc f ( ) sc t f ( ) (sc )(sc ) + (t )(sc t ), f ( ) so! () L( ) (b) P ( ) + (c) Th grphs of th lir d qudrtic pproimtios fit th grph of th fuctio r.. f( ) t f ( ) sc f ( ) ( sc )(sc t ) ( ), so f! [, ] by [, ]. f( ) ( + ) k k f ( ) k( + ) k k f ( ) k( k )( + ) kk ( ), f ( ) k( k) so! kk ( ) P ( ) + k+ For k, w hv f( ) ( + ) d f ( ) 6. W my us th Rmidr Estimtio Thorm with, M 6, 6 d r, givig R ( ). (I this prticulr! cs it is ctully tru tht R ( ), sic f () is cubic polyomil.) Thus th bsolut rror is lss th whvr <.. I th itrvl [, ], this occurs wh <... Altrt mthod: Not tht P ( ) + +. Usig grphig tchiqus, ( + ) ( + + ) < wh <... Lt f( ). Th P ( ) + + +, so w my us 6 th Rmidr Estimtio Thorm with. Sic f ( ) ( ), which is o mor th. wh.,.. w my us M d r, givig R ( )! Thus, for., th mimum possibl bsolut rror is bout. (. ) Altrt mthod: Usig grphig tchiqus, wh., rror
21 Sctio Sic th Mcluri sris is , P ( ) () Sic f ( ) ( ), which is o mor th (. 9) wh., w my us M (. 9) d (. 9) r, givig R ( ). Thus,! 9. for., uppr boud for th mgitud of th + (b) l l( ) l( ) ( ) () pproimtio rror is b sf, uppr boud is. 7. Altrt mthod: Usig grphig tchiqus, wh., rror ( ).... Roudig up to Th sris pproimts t. (b). () No (b) Ys, sic dy d ( ) ( ) ( )!! + + ( ).!! Th costt trm of y is y(), d w my obti th rmiig trms of y by itgrtig th bov sris. + y ( ) ( + )! By substitutig for, th grl trm my lso b writt s ( ) ( )( )!. (c) Th powr sris quls th fuctio y for ll rl vlus of. This is bcus th sris for covrgs for ll rl vlus of, so Thorm of Sctio 9. implis tht th w sris lso covrgs for ll. 6. () Substitut for i th Mcluri sris for l( + ) giv t th d of Sctio 9.. l( ) Th sris pproimts sc. 8. Fls. If f ( ) hpps to b, th liriztio is costt fuctio. 9. Tru. Th cofficit of is f ( ).. D !!. E.. B.. A.. () si ( cos ) ( ) ( ) ( )!! 6! ( ) + ( ) ( )! i! i! i 6! i 8! i! 6 8 +, (b) drivtiv + 7 ( ) ( ) ( ) (c) prt (b) + si!! 7!
22 98 Sctio 9.. () It works. For mpl, lt. Th P. d P + si P.96, which is ccurt to mor th 6 dciml plcs. (b) Lt P π + whr is th rror i th origil stimt. Th P+ si P ( π + ) + si ( π + ) π + si But by th Rmidr Thorm, si <. 6 Thrfor, th diffrc btw th w stimt P+ si P d π is lss th. 6 iθ iθ 6. () + (cosθ+ i si θ) + (cos ( θ) + i si ( θ)) cosθ+ isiθ+ cosθisiθ cosθ cosθ 7. (b) iθ iθ (cosθ+ isi θ) (cos ( θ) + isi( θ)) i i (cosθ+ isi θ) (cosθisi θ) i i siθ siθ i d (cos b+ isi b) d ( )( b sib + bi cos b) + ( )(cosb + i sib) ( ) ( bi sib+ bicos b) + (cosb+ i si b) ( ) bi(cos b+ isi b) + (cos b+ isi b) ( + bi)( )(cosb+ isi b) + ( + bi) ( bi) 8. () Th drivtiv of th right-hd sid is bi ( + bi) ( + bi) + b ( bi) ( + bi) + b + b ( bi) ( bi), + b + + which cofirms th tidrivtiv formul. (b) cos bd+ i si bd + ( bi) d bi + + b ( bi) bi + b (cos b + i si b) ( cos b + b si b bi cos b + b + i si b) ( cosb+ bsi b) + b + i ( sib b cos b) Sprtig th rl d imgiry prts givs cos b d ( cos b+ bsi b) d + b si b d ( sib bcos b) + b Quick Quiz Sctios D. 6. A. + +!! + +. E.. () Sic th sris is gomtric, it covrgs if d oly if r <, whr r +. + < + < < <. Th itrvl of covrgc is (, ). (b) Th sris is gomtric with first trm d commo rtio r +. It thrfor covrgs to 6 +. Sctio 9. Rdius of Covrgc (pp. ) Eplortio Fiishig th Proof of th Rtio Tst. For L : lim + lim. + + For : L lim ( ) lim. ( + )
23 Sctio () d k lim l k k lim l. k (b) d k k k k + lim lim.. Figur 9. shows tht is grtr th d. Sic th itgrl divrgs, so must th sris. Figur 9.b shows tht is lss th + d. Sic th itgrl covrgs, so must th sris.. Ths two mpls prov tht L c b tru for ithr divrgt sris or covrgt sris. Th Rtio Tst itslf is thrfor icoclusiv wh L. Eplortio Rvisitig Mcluri Sris +. L lim i lim. Th sris + + covrgs bsolutly wh <, so th rdius of covrgc is.. Wh, th sris bcoms. Ech trm i this sris is th gtiv of th corrspodig trm i th divrgt sris of Figur 9.. Just s divrgs to +, this sris divrgs to.. Gomtriclly, w chrt th progrss of th prtil sums s i th figur blow: + + L tc. Quick Rviw 9.. lim lim + +. lim lim ( + ) +. lim! (Not: This limit is similr to th limit which is discussd t th d of Empl i Sctio 9..). lim ( + ) ( ) lim lim lim Sic > for 6,, b, d N Sic > for 6,, b, d N Sic > l for,, b l, d N. 9. Sic <! > d hc! for,, b N!, d.. Sic < > d hc for,, b, d N. Sctio 9. Erciss. 6 < < ; Th grph of th fuctio y + d P ( ) illustrts th support Th sris covrgs t th right-hd dpoit. As show i th pictur bov, th prtil sums r closig i o som limit L s thy oscillt lft d right by costtly dcrsig mouts.. W kow tht th sris dos ot covrg bsolutly t th right-hd dpoit, bcus divrgs (Eplortio of this sctio).
24 Sctio 9.. < < ; Th grph of th fuctio y.. illustrts th support. d P 9 ( ) ( ) ( )! +! d is th Tylor sris for!! which covrgs for ll. ( ) ( )! +! d is th Tylor sris for!! which covrgs for ll.. (cos )! + 6. (si )! + (cos ) d covrgs to.!!! (si ) d covrgs to.!!! 7. This is gomtric sris which covrgs oly for <, so th rdius of covrgc is. 8. This is gomtric sris which covrgs oly for + <, so th rdius of covrgc is. 9. This is gomtric sris which covrgs oly for ( + ) <, or + <, so th rdius of covrgc is lim lim i + Th sris covrgs for ( ) <, or <, d divrgs for > is., so th rdius of covrgc. This is gomtric sris which covrgs oly for <, or <, so th rdius of covrgc is. +. lim lim ( + ) + + i lim + Th sris covrgs for < d divrgs for >, so th rdius of covrgc is. +. lim lim ( + ) i lim Th sris covrgs for < d divrgs for >, so th rdius of covrgc is. + +!. lim lim lim ( + i )! + + Th sris covrgs for ll vlus of, so th rdius of covrgc is.. lim lim ( + ) lim i + Th sris covrgs for + < d divrgs for + >, so th rdius of covrgc is lim lim ( + ) + + [( + ) + ] i ( + ) lim Th sris covrgs for < d divrgs for >, so th rdius of covrgc is lim lim ( + )! +! lim ( + ) ( ) Th sris covrgs oly for, so th rdius of covrgc is lim lim i lim + Th sris covrgs for < d divrgs for >, so th rdius of covrgc is.
25 Sctio lim lim ( ) ( + ) + ( + ) lim Th sris covrgs for < > d divrgs for, so th rdius of covrgc is. + / +. lim lim i ( ) / + + lim ( ) ( ) Th sris covrgs for ( ) <, which is quivlt to <, or < d divrgs for >. Th rdius of covrgc is. + + π +. lim lim i + + π lim + π + π Th sris covrgs for + π < d divrgs for + π >, so th rdius of covrgc is. +. lim lim lim ( ) ( ) + i + + Th sris covrgs for ( ) <, which is quivlt to <, d divrgs for >. Th rdius of covrgc is.. This is gomtric sris with first trm d commo ( ) rtio r. It covrgs oly wh ( ) <, so th itrvl of covrgc is < <. Sum r ( ) ( ) + +. This is gomtric sris with first trm d commo ( + ) rtio r. It covrgs oly wh ( ) + <, so 9 9 th itrvl of covrgc is < <. Sum r ( + ) ( ) This is gomtric sris with first trm d commo rtio r. It covrgs oly wh th itrvl of covrgc is < < 6. Sum r <, so 6. This is gomtric sris with first trm d commo rtio r l. It covrgs oly wh l <, so th itrvl of covrgc is < <. Sum r l 7. This is gomtric sris with first trm d commo rtio. It covrgs oly wh <, so th itrvl of covrgc is < <. Sum r ( ) 8. This is gomtric sris with first trm d commo rtio si. Sic si covrgc is < <. Sum r si si < for ll, th itrvl of
26 Sctio Divrgs by th th-trm Tst, sic lim. +. Divrgs by th th-trm Tst, sic lim. + (Th Rtio Tst c lso b usd.). Covrgs by th Rtio Tst, sic lim lim ( ) < i.. Covrgs, bcus it is gomtric sris with r 8, so r <.. Covrgs by th Rtio Tst, sic lim lim i ( + ) <. Altrtly, ot + tht + < for ll. Sic covrgs, covrgs by th Dirct Compriso Tst. +. Divrgs by th th-trm Tst, sic lim si. Covrgs by th Rtio Tst, sic lim lim ( ) + + <. 6. Covrgs by th Rtio Tst, sic lim lim ( ) + + i < Covrgs by th Rtio Tst, sic lim lim ( )! + +!! i +!( + )! ( + )! + lim ( + ) <. 8. Divrgs by th th-trm Tst, sic lim Covrgs, bcus it is gomtric sris with r, so r <.. Divrgs by th Rtio Tst, sic + lim lim ( + )! lim ( + ).! (Th th-trm Tst c lso b usd.). Divrgs by th Rtio Tst, sic + + lim lim i ( ) + + lim ( ) + ( ) >. (Th th Trm Tst c lso b usd.). Covrgs by th Rtio Tst, sic lim lim ( )l( ) i + l + l( + ) lim i i l <.. Covrgs by th Rtio Tst, sic + lim lim ( + )! ( + )! i ( + )!! + lim ( + )( + ) lim <. ( + ). Covrgs by th Rtio Tst, sic lim lim ( )! + + i + ( + )! ( + ) lim ( + )( + ) lim + lim ( + / ) <. O possibl swr: divrgs (s Eplortio i this sctio) v though lim. 6. O possibl swr: Lt d b Th db r covrgt gomtric sris, but b is divrgt gomtric sris. 7. Almost, but th Rtio Tst wo t dtrmi whthr thr is covrgc or divrgc t th dpoits of th itrvl.
27 Sctio ( )( ) + + s s s s + S lim s 6 9. ( )( ) + + s s ( )+ s ( ) s + S lim s. ( ) ( + ) ( ) ( + ) s 9 s s s ( + ) S lim s ( ) ( ) s s s s ( + ) S lim s. s s + s + + s + S lim s. s l l s + l l l l l l + s l l l l + l l l l s l( + ) l S lim s l π. s t t t s (t t ) + (t t ) π t s (t t ) + (t t ) + (t t ) π t π s t ( + ) π S lim s lim t. Tru. S Thorm Fls. Th powr sris π π π c ( ) lwys covrgs t. 7. B. 8. C. ( ) + ( ) ( ) E.
28 Sctio D. ( )( + ) ( ) ( + ) ( + ) lim ( + ) 6. () For k N, it s obvious tht c. k N N+ For ll k > N, c + c k N N k N N+ k + + c N N+ (b) Sic ll of th r ogtiv, th prtil sums of th sris form odcrsig squc of rl umbrs. Prt () shows tht th squc is boudd bov, so it must covrg to limit. 6. () For k N, it s obvious tht d + d d + d +. k N N+ For ll k > N, d+ d d+ d + d + + d d + d + + k N N k N N+ k d + d + N N+ (b) If covrgd, tht would imply tht d ws lso covrgt. 6. Aswrs will vry. 6. Diffrtit: ( ) Multiply by : ( ) Diffrtit: d d ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) + ( ) + ( ) + ( ) Multiply by : ( + ) ( ) Lt : 6 Th sum is 6. Sctio 9. Tstig Covrgc t Edpoits (pp. ) Eplortio Th p -SrisTst. W first ot tht th Itgrl Tst pplis to y sris of th form whr p is positiv. This is bcus th p fuctio f( ) p is cotiuous d positiv for ll >, d f ( ) p i p is gtiv for ll >. If p > : k d k p+ d lim p lim k p k p + lim i k p p k + ( sic p > ) p <. p Th sris covrgs by th Itgrl Tst.. If < p < : d k lim d p k p k p+ lim k p + p lim k k i ( ) p ( sic p > ). Th sris divrgs by th Itgrl Tst. If p, th sris divrgs by th th-trm Tst. This complts th proof for p <.
29 Sctio 9.. If p : d k d p k k lim ( l k ) lim l k. lim k Th sris divrgs by th Itgrl Tst. Eplortio Th Mcluri Sris of Strg Fuctio. Sic f ( ) ( ) for ll, th Mcluri Sris for f hs ll zro cofficits! Th sris is simply i.. Th sris covrgs (to ) for ll vlus of.. Sic f ( ) oly t, th oly plc tht this sris ctully covrgs to its f-vlu is t. Quick Rviw 9.. Covrgs, sic it is of th form d with p >. p. Divrgs, limit compriso tst with itgrl of.. Divrgs, compriso tst with itgrl of.. Covrgs, compriso tst with itgrl of. Divrgs, limit compriso tst with itgrl of 6. Ys, for N. 7. Ys, for N. 8. No, ithr positiv or dcrsig for >. 9. No, oscillts.. No, ot positiv for. Sctio 9. Erciss. f( ) / d ( ) / divrgs.. f( ) / d / / covrgs.... S, S, S, S S, S 6 b. K. Compr with d l divrgs. 6. Compr with d l( ) covrgs. 7. Divrgs by th Itgrl Tst, sic d divrgs Divrgs bcus p-sris Tst. /, which divrgs by th 9. Divrgs by th Dirct Compriso Tst, sic l for d divrgs. >. Divrgs by th Itgrl Tst, sic d divrgs.. Divrgs, sic it is gomtric sris with r l... Covrgs, sic it is gomtric sris with r 9.. l. Divrgs by th th-trm Tst, sic lim si.. Covrgs by th Dirct Compriso Tst, sic < for, d covrgs s + gomtric sris with r 7... Covrgs by th Dirct Compriso Tst, sic + < for, d / covrgs s / p-sris with p.
30 6 Sctio Covrgs by th Limit Compriso Tst, sic lim ( + )( + ), d covrgs s p-sris with p. 7. Divrgs by th th-trm Tst, sic + lim. 8. Covrgs by th Altrtig Sris Tst. If u l, th { u } is dcrsig squc of positiv trms with u lim. 9. Divrgs by th th-trm Tst, sic lim.. Covrgs by th Altrtig Sris Tst. If u + +, th u u { } is dcrsig squc of positiv trms with lim. (To show tht u is dcrsig, lt f( ) + d obsrv tht + + ( ) ( + )( ) f ( ), ( + ) ( +) which is gtiv, t lst for.). Divrgs by th th-trm Tst, sic l l which ms ch trm is ±. l, l. Divrgs by th Limit Compriso Tst. Lt d b. Th > d b > for d lim lim lim lim b. Sic b divrgs, lso divrgs.. Covrgs coditiolly: If u + +, th{ u } is dcrsig squc of positiv trms with lim u, so ( ) covrgs by th Altrtig Sris Tst But divrgs by th Dirct Compriso Tst, sic + for d divrgs.. Covrgs bsolutly, bcus, bsolutly, it is gomtric sris with r... Covrgs coditiolly., th { u } is dcrsig squc of If u l + positiv trms with lim u, so ( ) l covrgs by th Altrtig Sris Tst. But divrgs by th itgrl tst, sic l d b lim l l. l b 6. Covrgs bsolutly, sic covrgs by th Rtio Tst: + + lim ( + ) i <. 7. Divrgs by th th-trm Tst, sic lim! d so th trms do ot pproch. si 8. Covrgs bsolutly, sic covrgs by dirct compriso to, which covrgs s p-sris with p. 9. Covrgs coditiolly: If u, th { u } is dcrsig squc of + ( ) positiv trms with lim u, so covrgs by + th Altrtig Sris Tst. But divrgs by dirct compriso to, / + which divrgs s p-sris with p.. Covrgs bsolutly, sic covrgs s p-sris. cos π, which /
31 Sctio cos π ( ). Covrgs coditiolly, sic (S Empls d.). Covrgs codtiolly: If u + + ( ) of positiv trms with lim u, so + + covrgs by th Altrtig Sris Tst. But If v u Sic, th { u } is dcrsig squc divrgs by th Limit Compriso Tst: / u v + +, th lim lim. / v u divrgs s p-sris with p, lso divrgs Th positiv trms divrg ( + ) to d th gtiv trms 7 + divrg to. Aswrs 6 ( ) will vry. Hr is o possibility. () Add positiv trms util th prtil sum is grtr th A. Th dd gtiv trms util th prtil sum is lss th. Th dd positiv trms util th prtil sum is grtr th. Th dd gtiv trms util th prtil sum is lss th. Rpt this procss so tht th prtil sums swig rbitrrily fr i both dirctios. (b) Add positiv trms util th prtil sum is grtr th. Th dd gtiv trms util th prtil sum is lss th. Cotiu i this mr idfiitly, lwys closig i o.. Th positiv trms l l 7l 7 ( + )l( + ) + divrg to d th gtiv trms divrg to l l 6l 6 ( )l( ). Aswrs will vry. Hr is o possibility. () Add positiv trms util th prtil sum is grtr th. Th dd gtiv trms util th prtil sum is lss th. Th dd positiv trms util th prtil sum is grtr th. Th dd gtiv trms util th prtil sum is lss th. Rpt this procss so tht th prtil sums swig rbitrrily fr i both dirctios.. (b) Add positiv trms util th prtil sum is grtr th. Th dd gtiv trms util th prtil sum is lss th. Cotiu i this mr idfiitly, lwys closig i o.. This is gomtric sris which covrgs oly for <. () (, ) (b) (, ) (c) No 6. This is gomtric sris which covrgs oly for + <, or 6< <. () (6, ) (b) (6, ) (c) No 7. This is gomtric sris which covrgs oly for + <, or < <. () (b) (c) No,, lim lim + Th sris covrgs bsolutly wh <, or ( ) < <. Chck : covrgs coditiolly. Chck : divrgs. (), (b), (c) At 9. This is gomtric sris which covrgs oly for <, or 8< <. () (8, ) (b) (8, ) (c) No
32 8 Sctio 9.. lim lim ( + ) i Th sris covrgs bsolutly wh <, or < <. For, th sris divrgs by th th-trm Tst. () (, ) (b) (, ) (c) No +. lim lim ( + ) + i + + i Th sris covrgs bsolutly for <. Furthrmor, wh, which lso c, ovrgs s p-sris with p. () [, ] (b) [, ] (c) No + +!. lim lim lim ( + i )! + + Th sris covrgs bsolutly for ll rl umbrs. () All rl umbrs (b) All rl umbrs (c) No +. lim lim ( ) i + + Th sris covrgs bsolutly for + < < +, or 8 <. For, th sris divrgs by th th-trm Tst. () (8, ) (b) (8, ) (c) No +. lim lim ( + ) + + ( + ) + + i ( ) Th sris covrgs bsolutly for <, or < <. Chck : ( ) covrgs by th Altrtig Sris Tst. + Chck : divrgs by th Limit Compriso Tst with +. () [, ) (b) (, ) (c) At +. lim lim i Th sris covrgs bsolutly for <, or < <. For, th sris divrgs by th th-trm Tst. () (, ) (b) (, ) (c) No 6. lim lim ( + )! +! lim ( + ),, () Oly t (b) At (c) No ( + ) + 7. lim lim ( + ) ( ) Th sris covrgs bsolutly for ( ) <, or < <. For ( ), th sris divrgs by th th- Trm Tst. (), (b), (c) No
33 Sctio lim lim i ( + ) ( ) + Th sris covrgs bsolutly for ( ) <, or < <. Chck : + ( ) covrgs s p-sris with p. Chck : / covrgs s p-sris with p. (), (b), (c) No + + π + 9. lim lim i + + π Th sris covrgs bsolutly for + π <, or π < < π +. Chck π : + π ( ) covrgs by Altrtig Sris Tst. Chck π + : divrgs s p-sris with p. () [ π, π + ) (b) ( π, π + ) (c) At π. This is gomtric sris which covrgs oly for l <, or < <. (), (b), (c) No 9 7. i 6 i i l( + ) < sum < + l 7 7 l( ) < sum < + l( ). 8 < sum <. 8. < sum <.. Comprig rs i th figurs, w hv for ll +, f( ) d < + < + f( ) d. If th itgrl divrgs, it must go to ifiity, d th first iqulity forcs th prtil sums of th sris to go to ifiity s wll, so th sris is divrgt. If th itgrl covrgs, th th scod iqulity puts uppr boud o th prtil sums of th sris, d sic thy r odcrsig squc, thy must covrg to fiit sum for th sris. (S th pltio prcdig Ercis i Sctio 9..). y y y f () N N+ N N + N+ + y f () N N N + N+ Comprig rs i th figurs, w hv for ll + N, f ( ) d < N + < N f d N + ( ). N If th itgrl divrgs, it must go to ifiity, d th first iqulity forcs th prtil sums of th sris to go to ifiity s wll, so th sris is divrgt. If th itgrl covrgs, th th scod iqulity puts uppr boud o th prtil sums of th sris, d sic thy r odcrsig squc, thy must covrg to fiit sum for th sris. (S th pltio prcdig Ercis i Sctio 9..). () Divrgs by th Limit Compriso Tst. Lt k d bk. Th k > / k + 7 k k k d bk > for k d lim lim. k b k k + 7 Sicbk divrgs s p-sris with p k, k lso k divrgs. (b) Divrgs by th th-trm Tst, sic lim +. k k k (c) Covrgs bsolutly by th Compriso Tst, sic cos k < for k d covrgs k + k k k k s p-sris with p. (d) Divrgs by th itgrl tst, sic 8 8 d b lim l l l b k
34 Sctio 9. ( + ).! + ( + )! lim ( + ) ( + ) lim ( + ) ( + ) + < <, or. < < ! 6. lim ( ) + + +! + ( + )! lim ( ) + < < or. 9 < <. 9 ( + )! 7. O possibl swr: l This sris divrgs by th itgrl tst, sic d b lim l l. l Its prtil sums r b roughly l(l ), so thy r much smllr th th prtil sums for th hrmoic sris, which r bout l. 8. () ( ) 6( k) d k k+ k k+ k ( ) k k+ ( ) k (b) Th sris covrgs by th Altrtig Sris Tst. (c) Th first fw prtil sums r: S, S, S, S, S, S6, S7, S8, S9. For ltrtig sris, 6 th sum is btw y two djct prtil sums, so < S8 sum S9 <. 9. () Divrgs by th Limit Compriso Tst. Lt d b. Th > d b > for +, d lim lim Sic b +. b divrgs, divrgs. (b) S i. + + This sris covrgs by th Dirct Compriso Tst, sic + < d is covrgt s p-sris with p. 6. () From th list of Mcluri sris i Sctio 9., + l( + ) + + ( ). (b) < (c) To stimt l, w would lt Th tructio rror is lss th th mgitud of th sith ozro trm, or 6 < i 6 8 Thus, boud for th (bsolut) tructio rror is ( ) ( ) ( ) (d) l( + ) k+ k+ k+ l ( k + ) 6. lim lim i k k k l ( k + ) k k Th sris covrgs bsolutly for < < <, or. Chck : k k ( ) covrgs by th Altrtig Sris Tst. l( k + ) Chck : k l( k + ) divrgs by th Dirct Compriso Tst, sic > for k d divrgs. Th l( k+ ) k k k origil sris covrgs for <.
35 Sctio () Th sris covrgs by th Dirct Compriso Tst, sic < for, d covrgs s p l p p p-sris wh p >. (b) For p, th sris is, which divrgs by l p th Itgrl Tst, sic d b lim l(l ). l b (c) For p <, w hv > so p p l l, l divrgs by th Dirct Compriso Tst with from prt (b). l + 6. l( ) ( ) +, so t, th sris is + ( ). This sris covrgs by th Altrtig Sris Tst rct ( ) + At, th squc is + ( ) ( ) ( ), which covrgs by th + + Altrtig Sris Tst. At, th squc is ( ), which covrgs by th Altrtig Sris + Tst. 6. () It fils to stisfy u u for ll N. (b) Th sum is + / / / /. 66. Tru. Th dpoits r ±. Th corrspodig sris + ( ) is t ch dpoit, d it covrgs. 67. Tru. ( ) is gtiv. ( ) 68. B. ( )( ) ( ) ( + + )( ) + /, R / 69. A. Sic / d R,th / itrvl of covrgc is < <. 7. E. 7. C. ( ) Aswrs will vry. 7. () lim lim lim ( ) covrgs. Th sris (b) lim lim lim covrgs. (c) lim lim, odd, odd lim, v, Thus, lim lim, odd lim v, so th sris covrgs. 7. () lim lim Th sris covrgs bsolutly if <, or < <. Chck : ( ) divrgs. Chck : divrgs. Th itrvl of covrgc is (, ). Th sris (b) lim lim lim Th i i sris covrgs bsolutly if <, or < <. Chck : ( ) covrgs. Chck : divrgs. Th itrvl of covrgc is [, ).
36 Chptr 9 Rviw 7. Cotiud (c) lim lim Th sris covrgs bsolutly if <, or < <. Chck : ( ) divrgs. Chck : divrgs. Th itrvl of covrgc is,. (d) lim lim l l Th sris covrgs bsolutly if l <, or < <. Chck: : l ( ) divrgs. Chck : ( l ) divrgs. Th itrvl of covrgc is,. Quick Quiz Sctios 9. d 9.. E.. E. S. D. S () Rtio tst: lim + + lim ( + ) + + i + + lim ( + ) ( + ) i + + ( + )( ) + < < < Th sris covrgs bsolutly o (, ). (b) Th sris divrgs t both dpoits by th th-trm Tst: lim ( ( ) ) lim ( ( ) ) + + d. + + Sic th sris covrgs bsolutly o (, ) d divrgs t both dpoits, thr r o vlus of for which th sris covrgs coditiolly. Chptr 9 Rviw Erciss (pp. 6 9) + +!. lim lim lim ( + i )! + Th sris covrgs bsolutly for ll. () (b) All rl umbrs (c) All rl umbrs (d) No lim lim i ( ) Th sris covrgs bsolutly for + <, or 7< <. Chck 7: ( ) covrgs. Chck : divrgs. () (b) [7, ) (c) (7, ) (d) At 7. This is gomtric sris, so it covrgs bsolutly wh r < d divrgs for ll othr vlus of. Sic r ( ), th sris covrgs bsolutly wh ( ) <, or < <. () (b) (c) (d) No,, + ( )!. lim lim i ( + )! lim ( + )( ) Th sris covrgs bsolutly for ll. () (b) All rl umbrs (c) All rl umbrs (d) No
37 Chptr 9 Rviw + +. lim lim i ( + ) Th sris covrgs bsolutly for <, or < <. Furthrmor, wh, w hv d covrgs s p-sris with p,so lso covrgs bsolutly t th itrvl dpoits. () (b), (c), (d) No 6. lim lim ( + ) + ( + ) + Th sris covrgs bsolutly for <, or < <. Wh, th sris divrgs by th th-trm Tst. () (b) (, ) (c) (, ) (d) No + 7. lim lim ( + ) + + ( + ) i + ( + ) ( + ) + + Th sris covrgs bsolutly for + <, or < <. Wh +, th sris divrgs by th th-trm Tst. () (b), (c), (d) No lim lim i lim ( ) + + ( + )( + ) lim lim + Th ( + ) + sris covrgs bsolutly for ll. Aothr wy to s tht th sris must covrg is to obsrv tht for, w hv, so th trms r (vtully) boudd by th trms of covrgt gomtric sris. A third wy to solv this rcis is to us th th-root Tst (s Erciss 7 8 i Sctio 9.). () (b) All rl umbrs (c) All rl umbrs (d) No lim lim i Th sris covrgs + bsolutly for <, or < <. Chck : ( ) covrgs by th Altrtig Sris Tst. Chck : divrgs s p-sris with p. () (b) [, ) (c) (, ) (d) At +. lim lim ( + ) + + i Th sris covrgs bsolutly for <, or < <. Furthrmor, wh, w hv d covrgs s p-sris with p,so lso covrgs bsolutly t th itrvl dpoits. ()
38 Chptr 9 Rviw. Cotiud (b), (c) (d) No, +. lim lim ( ) + + i + ( + ) sris covrgs bsolutly wh < <. Wh () (b) (, ) (c) (, ) (d) No +. lim lim + <, or Th, th sris divrgs by th th-trm Tst. + + i ( ) + Th sris covrgs bsolutly wh <, or < <. ( ) ( ) ( ) Chck : covrgs + + coditiolly by th Altrtig Sris Tst. ( ) Chck : covrgs coditiolly by th + Altrtig Sris Tst. () (b) [, ] (c) (, ) (d) At d +. lim lim ( )! + + i +! + lim ( ),, Th sris covrgs oly t. () (b) oly (c) (d) No + + l. lim lim i Th sris l( + ) covrgs bsolutly for <, or < <. ( ) Chck : covrgs by th Altrtig l Sris Tst. Chck : divrgs by th Dirct Compriso l Tst, sic () (b), (c), (d) At. lim lim ( + )! + ( + )! > for d divrgs. l + Th sris covrgs oly t. () (b) oly (c) (d) No lim( + ) ( ) 6. This is gomtric sris with r, so it covrgs bsolutly wh, or. It divrgs for ll othr vlus of. () (b) (, ) (c) (, ) (d) No 7. f( ) + + ( ) +, vlutd t +. Sum +.
39 Chptr 9 Rviw 8. f( ) l( + ) + + ( ), vlutd t. Sum + l l f( ) si ( )! +! + ( + +, )! vlutd t π. Sum si π.. f( ) cos ( )! +! + ( +, vlutd )! π π t. Sum cos.. f( ) + +! + +! +, vlutd t l l. Sum.. f( ) t + + ( ) + +, vlutd t π. Sum t. (Not tht 6 wh is rplcd by, th grl trm of t + bcoms ( ), which mtchs th grl trm giv i th rcis.). Rplc by 6 i th Mcluri sris for th d of Sctio ( 6) + ( 6) + + ( 6) ( 6) +. Rplc by i th Mcluri sris for th d of Sctio 9.. ( ) + ( ) + ( ) ( ) + + giv t giv t. Th Mcluri sris for polyomil is th polyomil itslf: ( ) Rplc by π i th Mcluri sris for si giv t th d of Sctio 9.. ( ) ( ) si ( ) ( ) + π π π π π + + +!! ( + )! 8. Rplc by i th Mcluri sris for si giv t th d of Sctio 9.. si +!! + ( ) + ( ) ( +)! si + + +!! 7!. + + ( ) + + ( + )! ( + )! ( ) +!! 7! ( + )! !! ( ) +!! !! ( )!. Rplc by i th Mcluri sris for cos giv t th d of Sctio 9.. cos ( ) ( ) ( ) ( + + )!! ( )! ( ) ( ) ( ) + + +!! ( )! +. Rplc by π i th Mcluri sris for giv t th d of Sctio 9.. π / π π π !! π π π !. Us th Mcluri sris for giv t th d of Sctio ( ) ( ) ( ) +!! ( ) +!!
40 6 Chptr 9 Rviw. Rplc by i th Mcluri sris for t giv t th d of Sctio 9.. t + ( ) ( ) + + ( ) ( ) + + ( ). Rplc by i th Mcluri sris for l + giv t th d of Sctio 9.. ( ) ( ) l( ) + ( ) + ( ) + 8 ( ). ( ) 6. Us th Mcluri sris for l + giv t th d of Sctio 9.. l( ) l + ( ) ( ) ( ) + ( ) + ( ) f ( ) ( ) f ( ) ( ) f ( ) ( ), so f ( )! f ( ) f ( ) 6( ) 6, so! ( ) ( ) f ( ) f ( )!( )!, so! ( ) ( ) ( ) ( ) f( ) ( + ) f ( ) ( ) 7 f ( ) f ( ) ( 6 ), so! f ( ) f ( ) 6 6, so! ( f ) ( ) for ( + ) ( + ) + ( + ) This is fiit sris d th grl trm for is. 9. f () f () 9 f () f (), so 7! 7 f () f () 6, so 7! 8 ( ) () f ( )! + ( ) + ( ) ( ) ( ) + ( ) +. f( π) si π f ( π) cos π f ( π) f ( π) si π, so! f ( π) f ( π) cos π, so! 6, if k is v ( k) f ( π ), if k +, v, if k +, odd si ( π ) +! ( π)! ( π ) 7 + 7! ( π ) + + ( ) ( + )! ( π) + +. Divrgs, bcus it is tims th hrmoic sris:. Covrgs coditiolly. If u, th u { } is dcrsig squc of positiv ( ) trms with lim u, so covrgs by th Altrtig Sris Tst. Th covrgc is coditiol bcus is divrgt p-sris p.. Covrgs bsolutly by th Dirct Compriso Tst, sic l < for d covrgs s p-sris with p.. Covrgs bsolutly by th Rtio Tst, sic + +! lim lim lim ( + )! + + i. ( + ). Covrgs coditiolly: If u l( + ), th u { } is dcrsig squc of ( ) positiv trms with lim u, so covrgs l( + ) by th Altrtig Sris Tst. Th covrgc is coditiol bcus > for d l( + )
41 Chptr 9 Rviw 7. Cotiud divrgs, so l( + ) divrgs by th Dirct Compriso Tst. 6. Covrgs bsolutly by th Itgrl Tst, bcus d lim. (l ) b l l 7. Covrgs bsolutly th Rtio Tst, bcus + +! lim lim i ( + )! b lim Covrgs bsolutly by th Dirct Compriso Tst, sic for d is covrgt gomtric sris. Altrtly, w my us th Rtio Tst or th th-root Tst (s Ercis 7 d 8 i Sctio 9.). 9. Divrgs by th th-trm Tst, sic lim ( ) ( + ) + dos ot ist.. Covrgs bsolutly by th Dirct Compriso Tst, sic < d / covrgs s / ( + )( + ) p-sris with p.. Covrgs bsolutly by th Limit Compriso Tst. Lt d b. Th lim lim d covrgs s b p-sris (p ). Thrfor. Divrgs by th th-trm Tst, sic lim lim This is tlscopig sris. covrgs. ( )( ) ( ) ( ) s ( i ) ( i ) 6 s s s 6 ( ) S lim s 6. This is tlscopig sris ( ) s s + s s + + S lim s. () f () P f f () () + ()( ) + ( )! f ( + ) ( )! + ( ) + ( ) + ( ) f(.) P (.) 96. (b) Sic th Tylor sris for f c b obtid by trmby-trm diffrtitio of th Tylor Sris for f, th scod ordr Tylor polyomil for f t is + 6( ) + 6( ). Evlutd t.7, f ( 7. ) 7.. (c) It udrstimts th vlus, sic f ( ) 6, which ms th grph of f is cocv up r. 6. () Sic th costt trm is f (), f () 7. Sic f ( ), f ( ).! (b) Not tht P ( ) + ( ) 6( ) + ( ). Th scod dgr polyomil for f t is giv by th first thr trms of this prssio, mly + ( ) 6( ). Evlutig t., f (. ).. (c) Th fourth ordr Tylor polyomil for g() t is [ 7( t ) + ( t) ( t) ] d 7t ( t) + ( t) ( t) 7( ) ( ) + ( ) ( ) (d) No. O would d th tir Tylor sris for f (), d it would hv to covrg to f () t.
COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II
COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER 6 --- Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios.
More informationMAT 182: Calculus II Test on Chapter 9: Sequences and Infinite Series Take-Home Portion Solutions
MAT 8: Clculus II Tst o Chptr 9: qucs d Ifiit ris T-Hom Portio olutios. l l l l 0 0 L'Hôpitl's Rul 0 . Bgi by computig svrl prtil sums to dvlop pttr: 6 7 8 7 6 6 9 9 99 99 Th squc of prtil sums is s follows:,,,,,
More informationSOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3
SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos
More informationNational Quali cations
PRINT COPY OF BRAILLE Ntiol Quli ctios AH08 X747/77/ Mthmtics THURSDAY, MAY INSTRUCTIONS TO CANDIDATES Cdidts should tr thir surm, form(s), dt of birth, Scottish cdidt umbr d th m d Lvl of th subjct t
More informationNational Quali cations
Ntiol Quli ctios AH07 X77/77/ Mthmtics FRIDAY, 5 MAY 9:00 AM :00 NOON Totl mrks 00 Attmpt ALL qustios. You my us clcultor. Full crdit will b giv oly to solutios which coti pproprit workig. Stt th uits
More informationChapter 3 Fourier Series Representation of Periodic Signals
Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must:
More informationASSERTION AND REASON
ASSERTION AND REASON Som qustios (Assrtio Rso typ) r giv low. Ech qustio cotis Sttmt (Assrtio) d Sttmt (Rso). Ech qustio hs choics (A), (B), (C) d (D) out of which ONLY ONE is corrct. So slct th corrct
More informationLinear Algebra Existence of the determinant. Expansion according to a row.
Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit
More informationpage 11 equation (1.2-10c), break the bar over the right side in the middle
I. Corrctios Lst Updtd: Ju 00 Complx Vrils with Applictios, 3 rd ditio, A. Dvid Wusch First Pritig. A ook ought for My 007 will proly first pritig With Thks to Christi Hos of Swd pg qutio (.-0c), rk th
More informationENGI 3424 Appendix Formulæ Page A-01
ENGI 344 Appdix Formulæ g A-0 ENGI 344 Egirig Mthmtics ossibilitis or your Formul Shts You my slct itms rom this documt or plcmt o your ormul shts. Howvr, dsigig your ow ormul sht c b vlubl rvisio xrcis
More informationOption 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.
Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationIntegration by Guessing
Itgrtio y Gussig Th computtios i two stdrd itgrtio tchiqus, Sustitutio d Itgrtio y Prts, c strmlid y th Itgrtio y Gussig pproch. This mthod cosists of thr stps: Guss, Diffrtit to chck th guss, d th Adjust
More informationLE230: Numerical Technique In Electrical Engineering
LE30: Numricl Tchiqu I Elctricl Egirig Lctur : Itroductio to Numricl Mthods Wht r umricl mthods d why do w d thm? Cours outli. Numbr Rprsttio Flotig poit umbr Errors i umricl lysis Tylor Thorm My dvic
More informationHow much air is required by the people in this lecture theatre during this lecture?
3 NTEGRATON tgrtio is us to swr qustios rltig to Ar Volum Totl qutity such s: Wht is th wig r of Boig 747? How much will this yr projct cost? How much wtr os this rsrvoir hol? How much ir is rquir y th
More informationQuantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points)
Chm 5 Problm St # ANSWER KEY 5 qustios, poits Qutum Mchics & Spctroscopy Prof. Jso Goodpstr Du ridy, b. 6 S th lst pgs for possibly usful costts, qutios d itgrls. Ths will lso b icludd o our futur ms..
More information1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationQ.28 Q.29 Q.30. Q.31 Evaluate: ( log x ) Q.32 Evaluate: ( ) Q.33. Q.34 Evaluate: Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42. Q.43 Evaluate : ( x 2) Q.
LASS XII Q Evlut : Q sc Evlut c Q Evlut: ( ) Q Evlut: Q5 α Evlut: α Q Evlut: Q7 Evlut: { t (t sc )} / Q8 Evlut : ( )( ) Q9 Evlut: Q0 Evlut: Q Evlut : ( ) ( ) Q Evlut : / ( ) Q Evlut: / ( ) Q Evlut : )
More informationIX. Ordinary Differential Equations
IX. Orir Diffrtil Equtios A iffrtil qutio is qutio tht iclus t lst o rivtiv of uow fuctio. Ths qutios m iclu th uow fuctio s wll s ow fuctios of th sm vribl. Th rivtiv m b of orr thr m b svrl rivtivs prst.
More informationCLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC
CLSS XI ur I CHPTER.6. Proofs d Simpl pplictios of si d cosi formul Lt C b trigl. y gl w m t gl btw t sids d C wic lis btw 0 d 80. T gls d C r similrly dfid. T sids, C d C opposit to t vrtics C, d will
More informationterms of discrete sequences can only take values that are discrete as opposed to
Diol Bgyoko () OWER SERIES Diitio Sris lik ( ) r th sm o th trms o discrt sqc. Th trms o discrt sqcs c oly tk vls tht r discrt s opposd to cotios, i.., trms tht r sch tht th mric vls o two cosctivs os
More informationMath 1272 Solutions for Fall 2004 Final Exam
Mth 272 Solutios for Fll 2004 Fil Exm ) This itgrl pprs i Prolm of th udtd-2002? xm; solutio c foud i tht solutio st (B) 2) O of th first thigs tht should istigtd i plig th itgrtio of rtiol fuctio of polyomils
More informationReview Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
More informationFooling Newton s Method a) Find a formula for the Newton sequence, and verify that it converges to a nonzero of f. A Stirling-like Inequality
Foolig Nwto s Mthod a Fid a formla for th Nwto sqc, ad vrify that it covrgs to a ozro of f. ( si si + cos 4 4 3 4 8 8 bt f. b Fid a formla for f ( ad dtrmi its bhavior as. f ( cos si + as A Stirlig-li
More informationINTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)
Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More information. Determine these to one correct decimal accuracy using the bisection method: (a) 2. The following equations all have a root in the interval ( 0,1.
PROBLEMS Us grhic rrsttio to dtrmi th zros of th followig fuctios to o corrct dciml : ( 4 4si ; (b ; (c ( ; (d 4 8 ; ( ; (f ; (g t I ordr to obti grhicl solutio of f ( o th itrvl [,b], ty th followig sttmts
More informationMONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx
MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More information[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is
Discrt-tim ourir Trsform Rviw or discrt-tim priodic sigl x with priod, th ourir sris rprsttio is x + < > < > x, Rviw or discrt-tim LTI systm with priodic iput sigl, y H ( ) < > < > x H r rfrrd to s th
More information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
More informationSome Common Fixed Point Theorems for a Pair of Non expansive Mappings in Generalized Exponential Convex Metric Space
Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( 33-37 Som Commo Fi Poit Thorms for Pir of No psiv Mppigs i Grliz Epotil Cov Mtric Spc D.B.Oh Mish Kumr Mishr U Ktoch (Rsrch scholr Drvii Uivrsit
More informationH2 Mathematics Arithmetic & Geometric Series ( )
H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic
More informationTOPIC 5: INTEGRATION
TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function
More informationVtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya
LECTURE NOTES OF ENGINEERING MATHEMATICS III Su Cod: MAT) Vtusoutio.i COURSE CONTENT ) Numric Aysis ) Fourir Sris ) Fourir Trsforms & Z-trsforms ) Prti Diffrti Equtios 5) Lir Agr 6) Ccuus of Vritios Tt
More informationLectures 5-8: Fourier Series
cturs 5-8: Fourir Sris PHY6 Rfrcs Jord & Smith Ch.6, Bos Ch.7, Kryszig Ch. Som fu jv pplt dmostrtios r vilbl o th wb. Try puttig Fourir sris pplt ito Googl d lookig t th sits from jhu, Flstd d Mths Oli
More informationLectures 2 & 3 - Population ecology mathematics refresher
Lcturs & - Poultio cology mthmtics rrshr To s th mov ito vloig oultio mols, th olloig mthmtics crisht is suli I i out r mthmtics ttook! Eots logrithms i i q q q q q q ( tims) / c c c c ) ( ) ( Clculus
More informationThe z-transform. Dept. of Electronics Eng. -1- DH26029 Signals and Systems
0 Th -Trsform Dpt. of Elctroics Eg. -- DH609 Sigls d Systms 0. Th -Trsform Lplc trsform - for cotios tim sigl/systm -trsform - for discrt tim sigl/systm 0. Th -trsform For ipt y H H h with ω rl i.. DTFT
More informationCalculus Cheat Sheet. ( x) Relationship between the limit and one-sided limits. lim f ( x ) Does Not Exist
Clulus Cht Sht Limits Dfiitios Pris Dfiitio : W sy lim f L if Limit t Ifiity : W sy lim f L if w for vry ε > 0 thr is δ > 0 suh tht mk f ( ) s los to L s w wt y whvr 0 < < δ th f L < ε. tkig lrg ough positiv.
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationPURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationOptions: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
More information(HELD ON 22nd MAY SUNDAY 2016) MATHEMATICS CODE - 2 [PAPER -2]
QUESTION PAPER WITH SOLUTION OF JEE ADVANCED - 6 7. Lt P (HELD ON d MAY SUNDAY 6) FEEL THE POWER OF OUR KNOWLEDGE & EXPERIENCE Our Top clss IITi fculty tm promiss to giv you uthtic swr ky which will b
More informationA Simple Proof that e is Irrational
Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural
More informationIIT JEE MATHS MATRICES AND DETERMINANTS
IIT JEE MTHS MTRICES ND DETERMINNTS THIRUMURUGN.K PGT Mths IIT Trir 978757 Pg. Lt = 5, th () =, = () = -, = () =, = - (d) = -, = -. Lt sw smmtri mtri of odd th quls () () () - (d) o of ths. Th vlu of th
More informationPREPARATORY MATHEMATICS FOR ENGINEERS
CIVE 690 This qusti ppr csists f 6 pritd pgs, ch f which is idtifid by th Cd Numbr CIVE690 FORMULA SHEET ATTACHED UNIVERSITY OF LEEDS Jury 008 Emiti fr th dgr f BEg/ MEg Civil Egirig PREPARATORY MATHEMATICS
More informationCONTINUITY AND DIFFERENTIABILITY
MCD CONTINUITY AND DIFFERENTIABILITY NCERT Solvd mpls upto th sction 5 (Introduction) nd 5 (Continuity) : Empl : Chck th continuity of th function f givn by f() = + t = Empl : Emin whthr th function f
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More information1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.
Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b =
More informationTime : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120
Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More informationSection 3: Antiderivatives of Formulas
Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin
More informationMath 2414 Activity 17 (Due with Final Exam) Determine convergence or divergence of the following alternating series: a 3 5 2n 1 2n 1
Mth 44 Activity 7 (Due with Fil Exm) Determie covergece or divergece of the followig ltertig series: l 4 5 6 4 7 8 4 {Hit: Loo t 4 } {Hit: } 5 {Hit: AST or just chec out the prtil sums} {Hit: AST or just
More information+ x. x 2x. 12. dx. 24. dx + 1)
INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE INTEGRAL Fidig th idfiit itgrals Rductio to basic itgrals, usig th rul f ( ) f ( ) d =... ( ). ( )d. d. d ( ). d. d. d 7. d 8. d 9. d. d. d. d 9. d 9.
More informationERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**
ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults
More informationCalculus & analytic geometry
Calculus & aalytic gomtry B Sc MATHEMATICS Admissio owards IV SEMESTER CORE COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITYPO, MALAPPURAM, KERALA, INDIA 67 65 5 School of Distac
More informationIV. The z-transform and realization of digital filters
www.tuworld.com www.tuworld.com Digitl Sigl Procssig 4 Dcmbr 6, 9 IV. Th -trsform d rlitio of digitl filtrs 7 Syllbus: Rviw of -trsforms, Applictios of -trsforms, Solutio of diffrc qutios of digitl filtrs,
More informationLimits Indeterminate Forms and L Hospital s Rule
Limits Indtrmint Forms nd L Hospitl s Rul I Indtrmint Form o th Tp W hv prviousl studid its with th indtrmint orm s shown in th ollowin mpls: Empl : Empl : tn [Not: W us th ivn it ] Empl : 8 h 8 [Not:
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationStatistics 3858 : Likelihood Ratio for Exponential Distribution
Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai
More informationChapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1
Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd
More informationEmil Olteanu-The plane rotation operator as a matrix function THE PLANE ROTATION OPERATOR AS A MATRIX FUNCTION. by Emil Olteanu
Emil Oltu-Th pl rottio oprtor s mtri fuctio THE PLNE ROTTON OPERTOR S MTRX UNTON b Emil Oltu bstrct ormlism i mthmtics c offr m simplifictios, but it is istrumt which should b crfull trtd s it c sil crt
More informationChem 4502 Prof. Doreen Leopold 10/18/2017 Name (Please print) Quantum Chemistry and Spectroscopy Exam 2 (100 points, 50 minutes, 13 questions)
Chm 5 Prof. Dor Lopold ANSWER KEY /8/7 Nm Pls prit Qutum Chmistry d Spctroscopy Em poits, 5 miuts, qustios Pls chck hr if you would prfr your grdd m to b rturd to you dirctly rthr th big icludd mog lphbtizd
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationChapter 8 Approximation Methods, Hueckel Theory
Witr 3 Chm 356: Itroductory Qutum Mchics Chptr 8 Approimtio Mthods, ucl Thory... 8 Approimtio Mthods... 8 Th Lir Vritiol Pricipl... mpl Lir Vritios... 3 Chptr 8 Approimtio Mthods, ucl Thory Approimtio
More information( ) = A n + B ( ) + Bn
MATH 080 Test 3-SOLUTIONS Fll 04. Determie if the series is coverget or diverget. If it is coverget, fid its sum.. (7 poits) = + 3 + This is coverget geometric series where r = d
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationBC Calculus Path to a Five Problems
BC Clculus Pth to Five Problems # Topic Completed U -Substitutio Rule Itegrtio by Prts 3 Prtil Frctios 4 Improper Itegrls 5 Arc Legth 6 Euler s Method 7 Logistic Growth 8 Vectors & Prmetrics 9 Polr Grphig
More informationAn Introduction to Asymptotic Expansions
A Itroductio to Asmptotic Expasios R. Shaar Subramaia Asmptotic xpasios ar usd i aalsis to dscrib th bhavior of a fuctio i a limitig situatio. Wh a fuctio ( x, dpds o a small paramtr, ad th solutio of
More information(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is
[STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More informationTaylor and Maclaurin Series
Taylor ad Maclauri Sris Taylor ad Maclauri Sris Thory sctio which dals with th followig topics: - Th Sigma otatio for summatio. - Dfiitio of Taylor sris. - Commo Maclauri sris. - Taylor sris ad Itrval
More informationDiscrete Fourier Transform (DFT)
Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial
More informationk m The reason that his is very useful can be seen by examining the Taylor series expansion of some potential V(x) about a minimum point:
roic Oscilltor Pottil W r ow goig to stuy solutios to t TIS for vry usful ottil tt of t roic oscilltor. I clssicl cics tis is quivlt to t block srig robl or tt of t ulu (for sll oscilltios bot of wic r
More informationLectures 9 IIR Systems: First Order System
EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work
More informationSection 5.1/5.2: Areas and Distances the Definite Integral
Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto: Th Dt Itgrl Suppos w
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationdenominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationThe Propagation Series
/9/009 Th Progtio Sris.doc /8 Th Progtio Sris Q: You rlir sttd tht sigl flow grhs r hlful i (cout m ) thr wys. I ow udrstd th first wy: Wy - Sigl flow grhs rovid us with grhicl ms of solvig lrg systms
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationProblem Value Score Earned No/Wrong Rec -3 Total
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING ECE6 Fall Quiz # Writt Eam Novmr, NAME: Solutio Kys GT Usram: LAST FIRST.g., gtiit Rcitatio Sctio: Circl t dat & tim w your Rcitatio
More informationCh 1.2: Solutions of Some Differential Equations
Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of
More informationSTIRLING'S 1 FORMULA AND ITS APPLICATION
MAT-KOL (Baja Luka) XXIV ()(08) 57-64 http://wwwimviblorg/dmbl/dmblhtm DOI: 075/МК80057A ISSN 0354-6969 (o) ISSN 986-588 (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract:
More informationThe Propagation Series
//009 Th Progtio Sris rst /0 Th Progtio Sris Q: You rlir sttd tht sigl flow grhs r hlful i (cout m ) thr wys. I ow udrstd th first wy: Wy - Sigl flow grhs rovid us with grhicl ms of solvig lrg systms of
More informationEXERCISE - 01 CHECK YOUR GRASP
DEFNTE NTEGRATON EXERCSE - CHECK YOUR GRASP. ( ) d [ ] d [ ] d d ƒ( ) ƒ '( ) [ ] [ ] 8 5. ( cos )( c)d 8 ( cos )( c)d + 8 ( cos )( c) d 8 ( cos )( c) d sic + cos 8 is lwys posiiv f() d ( > ) ms f() is
More information2. Infinite Series 3. Power Series 4. Taylor Series 5. Review Questions and Exercises 6. Sequences and Series with Maple
Chpter II CALCULUS II.4 Sequeces d Series II.4 SEQUENCES AND SERIES Objectives: After the completio of this sectio the studet - should recll the defiitios of the covergece of sequece, d some limits; -
More informationTest Info. Test may change slightly.
9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow
More informationLaw of large numbers
Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs
More informationDTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
More informationMulti-Section Coupled Line Couplers
/0/009 MultiSction Coupld Lin Couplrs /8 Multi-Sction Coupld Lin Couplrs W cn dd multipl coupld lins in sris to incrs couplr ndwidth. Figur 7.5 (p. 6) An N-sction coupld lin l W typiclly dsign th couplr
More informationEEE 303: Signals and Linear Systems
33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d =
More informationImportant Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
More informationProblem Session (3) for Chapter 4 Signal Modeling
Pobm Sssio fo Cht Sig Modig Soutios to Pobms....5. d... Fid th Pdé oimtio of scod-od to sig tht is giv by [... ] T i.. d so o. I oth wods usig oimtio of th fom b b b H fid th cofficits b b b d. Soutio
More information