Math 2414 Activity 17 (Due with Final Exam) Determine convergence or divergence of the following alternating series: a 3 5 2n 1 2n 1

Transcription

1 Mth 44 Activity 7 (Due with Fil Exm) Determie covergece or divergece of the followig ltertig series: l {Hit: Loo t 4 } {Hit: } 5 {Hit: AST or just chec out the prtil sums} {Hit: AST or just chec out the prtil sums} 6 9 The ltertig series coverge to? coverges by the Altertig Series Test, but wht does it Let s loo t the eve prtil sums: s 4 4 Solvig the previous equtio for s, we get s s So lims lim

2 Fid lim, d you ll ow the sum of the series Activity, problem #0 Clculte lim by rewritig it s lim d idetifyig it s defiite itegrl Activity 4, problem # 4From the two grphs, you c coclude tht dx dx x x Use this fct to fid lim 0 Show tht i the Altertig Series Test, it is ot eough to hve lim 0 Cosider the ltertig series ; is odd 4 ; is eve ) Wht s lim?, where Cosider the eve prtil sums, s 4 9 b) Wht s lim s? c) Wht does this imply bout the covergece of ) Show tht?

3 b) Show tht if is sequece of positive umbers, the if l is decresig I other words, show tht if l l c) For 0 x, show tht l x x {Hit: l x is decresig, the, the x 0 dt, d t t } d) Show tht l is decresig sequece by showig tht f x xl l x hs egtive derivtive {Hit:Use prt c)} x e) Determie whether the ltertig series diverget usig the previous results is coverget or A perso strts wlig from home (t x 0) towrd fried s house (t x ) Threefourths of the wy there, he chges his mid d strts wlig bc home Three-fourths of the wy bc home, he chges his mid gi d strts wlig bc to his fried s house Three-fourths of the wy to where he first tured roud, he turs roud gi d strts wlig towrd home Three-fourths of the wy to where he tured roud the secod time, he turs roud gi d strts wlig towrd his fried s house If he cotiues this ptter idefiitely, where will he ed up? Here re his movemets: His fil positio will be the sum of ltertig series which meets the coditios of the Altertig Series Theorem, so he must ed up t some poit Fid it 0,, with f 0 Cosider the series Suppose tht f, f, d f re ll cotiuous o f ) If the series f is coverget, the wht must be the vlues of f 0 d 0 {Hit: f f, so use the th term test f tht 0 lim f 0 0 d use The Limit Compriso Test} 0 f? f f 0 lim, so suppose

4 b) If f 0 0 d f 0 0, c you coclude ythig bout the covergece or divergece of f? f f x {Hit: lim lim x0 x, provided tht lim x0 f x twice to get result i the (exteded)limit Compriso Test} 4 Determie if the improper itegrl si si Hit: Let u x d covert the itegrl ito 0 x exists Apply L Hopitl s Rule x x dx is coverget or diverget 0 si cosu u cosu u the boo, si usiu si u siu cosu u cosu u 0 0 usi u u du By trig idetity from, so we get tht du du u But 4 u cosu u cosu u cosu u cosu u cosu u du du du 4 u 4 u 4 u 0 0 cosu u du, 4 u provided tht ech itegrl coverges The first itegrl does, so let s loo t the itegrd cos x x of the secod oe: f x o the 4 x itervl, cos u u It ppers tht du is 4 u equivlet to ltertig series to which the Altertig Series Test c be pplied

5 cos u u Wht bout the secod itegrl du? 4 u 5 It c be show tht the ltertig series coverge to? coverges, but wht does it {Hit:, so use the result of problem #9} 6 The sequece lim x x x x is defied recursively by x0 0, x, d x ; Fid x x x x x x x x x x {Hit: 0 x x x x 4 So x x x x x x x x x0 ow let 0 } 7 The sequece lim x x x x is defied recursively by x0 0, x, d x ; Fid x x x x x x x x x x {Hit: x x x x0 x x x x x x x x x So 0 0 ow let }

6 8 Determie bsolute covergece, coditiol covergece, or divergece for the followig series: ) b) 9 The ltertig series is ot bsolutely coverget, but it is coditiolly coverget Show tht the followig rerrgemet of this series: hs differet sum Multiply the origil series by, isert term of 0 before ech of its terms, d the dd it to the origil series term by term From Problem #9, we ow tht Fid the vlue of the coverget ltertig series {Hit: ) If l Wht s the vlue of the rerrged series?, so, d use the result of Problem #9} b) Wht if is bsolutely coverget, the wht c be sid bout? is oly coditiolly coverget? {Hit: Thi bout }

7 Determie covergece or divergece for the followig series:!! {rtio test} {rtio test} 4! {rtio test} !! {rtio test} 7 e {rtio test} 8 0 r ; 0 r 9 7! 9 0 {rtio test}! {rtio test}!! Determie the covergece or divergece of the series recursively by the followig: si ; 4 ; 5 whose terms re defied 5 t ; 6 ; {Hit: Fid formul for } l 7 5; 8 ; 0 {Hit: Is icresig or decresig?}

8 ; {Hit: Fid formul for or } 4 ; 5 {Hit: Fid formul for or } ; {Hit: Determie lim by cob-web lysis} 4 Let d 4, {Hit: Determie lim d pply the Rtio Test}

9 4 For which positive itegers is the series!! coverget? 44 If coverges, the wht bout si? {Root Test} 45 I this problem, you ll exted the Root Test to series cotiig fctorils! Which implies tht! d! So you get! Which implies tht! Ad you lso get! Which implies tht! ) Determie lim! iequlities by cosiderig wht hppes for odd d eve itegers i the previous b) Apply the Root Test to the series!

10 46 Cosider the series, where ; is eve ; is odd ) Show tht the rtio test fils b) Wht hppes i the root test? 47 Remember from Activity 4, Problem #5, the Fibocci sequece F,,,,,5,8,,,4,55,89, F, determie if the series F coverges or diverges { F Assumig tht F, the F ;if F F So F for by iductio See if you c show tht F 5 for 5 by iductio} 48 Show tht the series coverges S which is prtil sum of ltertig series Sice y prtil sum of the origil series differs from multiple of prtil sum by t most terms which go to zero s teds to ifiity, if S is coverget the so is y prtil sum See if you c show tht 7 8 coverges

11 49 Show tht the series coverges d fid its sum S So S, but this mes tht 6 S, d so S Fid lim S 50 Cosider the series ) Determie if the series is coverget or diverget usig compriso {Hit: b) Determie if the series is coverget or diverget usig the limit of its prtil sums { } } 5 Is the ltertig series coverget or diverget? {otice tht you c t use the Altertig Series Test sice the s re t decresig}

12 5 Determie covergece or divergece of si si {Hit: The Me Vlue Theorem sys tht * x } * cos x si si for some 5 Cosider the series This series is rerrgemet of the coverget ltertig series This mes tht S for multiple of, stisfies rerrgemet of the coverget series S do? So wht does the i 54 Addig groupig symbols to coverget series does t chge its covergece or its sum We ow from Problem #9 tht l 4 5 ) By isertig groupig symbols ito the series 4 5 series 4 56, fid the sum of the b) Ad lso the sum of the series 45 67

13 55 Give,, d b,, b, let S b b b S b b S b b S b b S b or more Show tht i i i i i, which is commoly referred to s the i i succictly b S b S b b summtio by prts formul logous to the itegrtio by prts formul: i bi S b Si bi bi i dv u v u i v du For, the left side of the formul is b d the right side of the formul is b ow suppose the formul is true for, so we hve b b b S b b S b b S b b Sb, d let s dd b o both sides to get b b b b S b b S b b S b b S b b S b b S b b S b b S b b b S b See if you c show tht b Sb S b 56 Suppose tht b is o-egtive decresig sequece d i for ll i m M Show tht b m b b b b M for ll From problem #55, we ow tht b b b S b b S b b S b b S b m b b m b b mb b b b M b b M b b Mb m b b b b b b b b M b b b b b So simplify the cotets of the brcets to get the result 57 Suppose tht b is o-egtive decresig sequece d i for ll i m M Show tht for ech, b m b b b b M for ll From problem #55, we ow tht b b b S b b S b b S b b S b Fiish it off lie problem #56

14 58 (Dirichlet s Covergece Test) Suppose tht the prtil sums of the series bouded, ie Show tht the series for ll, d m M b coverges re b is decresig sequece with limb 0 To show tht series is coverget, it is eough to show tht the prtil sums of the series buch up, ie S S c be mde rbitrrily smll for m sufficietly lrge m i i m S S b, d from the previous problem, we ow tht im m i i m, d sice limb 0 im b m b b M choosig d m sufficietly lrge, we c me S S si Apply Dirichlet s Test to the series d cos {Hit: Let si, b, d cosider the sum si m rbitrrily smll by i From trigoometry, you ow tht e cos isi, so i i i i cos si 0i e e e e i, but lso from the geometric series e 0 0 i i i i e formul tht e e e e, i e cos i si i i i i cos isi so e e e e cos isi cos isi i cos cos coscos sisi cos si sicos si cossi i cos So puttig thigs together, we get tht

15 0 cos cos coscos sisi cos d tht cos 0 0 si sicos si cossi si So we hve tht cos 5 cos d cos si } cos 0 59 (Abel s Covergece Test) Suppose tht the series coverges d the sequece is either mootoe decresig or mootoe icresig d is bouded, ie ll Show tht the series {Hit: If b coverges b m b M for b is icresig, the it must hve limit, b, d the ew sequece decresig sequece whose limit is 0 Sice b b is coverges, its prtil sums must be bouded, so Dirichlet s Test pplied to the series b b coverges, d the series b b b b } Use Abel s Test so show tht if b c p coverges, the so does c q for q implies tht it {Hit: Let is coverget, so p c d p } b q p

16 Let s estblish coectio betwee the Rtio d Root tests Suppose tht 0 for ll d lim L for 0 L The lim l l l L, d for 0 we get tht, l L l l l L l L l l l L for So l L l m l m l L So ddig the iequlities, we get tht ml L l l ml L leds to l L l l l L m for m, which l L l m l l L, m m l l m l L l Sice 0 is rbitrry, it must be m m m m lim l m l L Sice l m l m, we lso m m m m m lim l m l L, d hece tht lim L If L 0, the for 0, m m for So we get tht 0 d to the cse tht get tht 0 0, d 0 m m m m Multiplyig the iequlities, we get 0, which implies tht m m m 0 So 0 m m m, d gi sice 0 is rbitrry, d tht m m m m m m, we get lim 0 L I the cse of L, the we hve tht for y B 0, B for m m As before, B, d hece, m

17 m m B So m m m B Sice B 0 is rbitrry, d m m m m it must be the cse tht lim L This mes tht if lim L, the lim L d therefore, coclusio bout series bsed o the rtio test will lwys yield the sme coclusio from the root test However, the root test my yield result while the rtio test fils to yield result Use the previous discussio to fid the followig limits: m m 60 lim! {Hit: Let d loo t the rtio }! 6 lim! {Hit: Let! d loo t the rtio } 6 lim {Hit: Let d loo t the rtio }

18 Kummer s Test: Suppose tht, b 0, b, d lim L If L 0, the coverges, d if b b L 0, the diverges Cse I: L 0 L L The for,, which implies tht or b b b b S M so is, but this mes tht L b b M M L b b L b b L b M So M Cse II: L 0 The for b, b is coverget, d 0, which implies tht b, so we get tht b b b b b b b b b b b b b b b b b b b b b b b M M b M M M M bm bm b So M M b b, therefore d re diverget Rbe s Test is specil cse of Kummer s Test tht is esier to pply:

19 Rbe s Test: Suppose tht 0 coverges, d if L 0, the d lim diverges L If L 0, the 6 Show tht Rbe s Test follows from Kummer s Test {Hit: Let b i Kummer s Test} 64 Use Rbe s Test o the series 65 Use Rbe s Test o the series 66 Use Rbe s Test o the series !! 67 ) Let s estimte the error for prtil sum of series for which the rtio test pplies Cosider the series R If S {Hit: R for, the show tht R i b) Estimte the error, from prt ), i usig the 0 th prtil sum of the series i {Hit: 0 } for 0}

20 68 ) Let s estimte the error for prtil sum of series for which the root test pplies Cosider the series R If S {Hit: R for, the show tht R i b) Estimte the error, from prt ), i usig the 0 th prtil sum of the series 69 Determie if the ltertig series groupig the terms i pirs {Hit: {Hit: i! } for } coverges by } 70 Do the sme for the series {Hit: } b b 7 Cosider the ltertig series for b, 0 4 ) Determie covergece or divergece if b b b 4 } {Hit:

21 b) Determie covergece or divergece if b Suppose tht b, the b b b b b b b 4 4 Fiish it from here 7 Try the Rtio Test o the series other method Determie covergece or divergece with 7 Try the Root Test o the series with other method 5 Determie covergece or divergece 74 There is extesio of the Rtio Test clled the Secod Rtio Test Here s wht it sys: Secod Rtio Test: Let be series with positive terms, d suppose tht lim lim both exist Let L mx lim,lim l mi lim,lim The i) If ii) If L, the l, the coverges diverges iii) If l L, the the test fils ) If you ttempt the Rtio Test o the series, you do t get result Use the Secod Rtio Test o it b) If you ttempt the Rtio Test o the series, you do t get result Use the Secod Rtio Test o it 5 c) If you ttempt the Rtio Test o the series, you do t get result! Use the Secod Rtio Test o it d d

22 75 Cosider the ltertig series ) u Show tht 0 b) Show tht u for ll, d limu 0, d hece tht c) Is the ltertig series coverget or diverget? 76 ) For p, fid simple formul for the vlue of f p coverget p p p 4 p p p 4 p p diverget {Hit: p p p p p p p p p } lim f p b) Fid p f p for p? 77 Let s use the Altertig Series Test Estimtio to show tht rerrgemet of the Altertig Hrmoic Series coverges to differet limit We ll strt with the Altertig Hrmoic Series i stdrd order: 4 c) Wht s wrog with ) Use the first three terms to show tht the sum of the Altertig Hrmoic Series i stdrd order is less th 5 6 Cosider the rerrgemet of the Altertig Hrmoic Series: Addig groupig symbols results i other ltertig series:

23 ; eve This ew ltertig series c be writte s, with 4 ; odd 4 b) The ew ltertig series stisfies the coditios of the Altertig Series Test d hece coverges Use the first two terms to show tht its sum is greter th 5 6, d hece differet th the origil sum 78 C series both coverge d diverge? Cosider the ltertig series l O the other hd, It c be show to be coverget usig the Altertig Series Test l l l l But l l l l l l l 4 l l5 l 4 ppers tht So it l l l l l 4 l 5 l 6 l, but l diverges by the th term test Wht hppeed? {Uder wht coditios c pretheses be dded or removed from series without ffectig its covergece? For exmple,? ? d How do the prtil sums of the left-hd d right-hd sides compre i both situtios?} 79 It c be show tht the ltertig hrmoic series coverges Suppose tht S The S 4 5 6, d rerrgig d S groupig leds to Dividig S both sides by S, leds to the flse equtio Did somethig go wrog? Expli

24 80 Cosider the series si test, but you d be miste You might thi tht it diverges by the th term From the Me Vlue Theorem, we get tht cos c si si si, where mes tht 0 limsi 0 c c cos c, or 0 si c This So, which mes tht we c t coclude divergece But the Me Vlue Theorem result bove tells us tht si c Ad cos c c cos c, where 0, which mes tht bsolutely coverget So ow we ow tht si 4 pproch with the series si c cos is c c coverges Try similr cos 4 c, {Hit: Show tht si 4 si 4 si 4 8 Suppose tht, d ; coverges or diverges where 4 c c } 4 4 Determie if the series {Hit: Exmie the sequece } 8 You might thi tht if b is diverget series, d be diverget series Show why you would be miste b, the would hve to {Let b,, d see wht hppes}

25 8 You might thi tht if b is diverget series, d b be diverget series Show why you would be miste {Let b 84 You might thi tht if, coverges, d limc 0 coverge Show why you would be miste 85 You might thi tht if b {Let, c, the would hve to, d see wht hppes}, the c coverges, d lim, the b would hve to, d see wht hppes} would hve to coverge Show why you would be miste {Let b,, d see wht hppes}