Chapter 9 Infinite Series

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.