Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
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1 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 tst F Imror Itgrals To aly th itgral tst you d to b familiar with itgratio i articular imror itgrals. W will b usig imror itgrals throughout this sctio. What dos th trm imror itgral ma? A itgral i which o or both of th limits of itgratio is ifiit. A aml is f ( d which is ormally valuatd by lim f ( d whr > If this is a fiit valu th th limit ists ad w say th itgral is covrgt. This is th sam trm as w us i this chatr for ifiit sris. If th limit dos ot ist th what ca w say about th abov itgral? Th itgral is divrgt that is if th imror itgral dos ot hav a fiit valu th it divrgs. Dfiitio (.5. Lt f : b a fuctio dfid for a whr a is a ral umbr. Th th imror itgral is dfid by f ( d lim f ( d whr > a a If this itgral has a fiit valu th th imror itgral ists ad w say th itgral covrgs. If th limit dos ot ist th th imror itgral divrgs. Thr is aothr ty of imror itgral that is wh th itgrad is ifiit withi th rgio of itgratio but w will oly coctrat o th imror itgral dfid i (.5. Lt s do a aml. Eaml 3 Dtrmi whthr th followig imror itgral ( covrgs. If it dos covrg dtrmi its valu. Solutio. Lt ad b ositiv th by dfiitio (.5 abov w hav ( d lim ( d (* Lt s ami th dfiit itgral i (* d
2 Chatr Ifiit Sris Pag of ( ( Bcaus ( d d Substitutig this, ( Why is ( lim? Bcaus ( Substitutig Limits ad Takig out th Ngativ Sig ( Takig i th Ngativ Sig d lim lim, ito (* givs ( d lim ( ( lim, sic as thrfor ( Hc th giv imror itgral d covrgs with a valu of.. Eaml 3 Dtrmi whthr th followig imror itgral covrgs d If it dos covrg dtrmi its valu. Solutio. Lt ad b ositiv th by dfiitio (.5 w hav d d lim ( d Lt s ami th itgral i ( : d d l Bcaus l ( ( l ( l ( l ( Substitutig this rsult d l ( ito ( givs Hc th giv imror itgral, d lim ( l ( + d, divrgs.
3 Chatr Ifiit Sris Pag 3 of Why is th toic imror itgrals i this chatr o ifiit sris? Th aswr to this qustio is giv i th t sctio. F Harmoic Sris ad th Itgral Tst What is th ara udr th curv from to + qual to? This is th itgral giv by d Rmmbr th (dfiit itgral givs th ara udr th curv. W ca also aroimat th ara udr th curv f ( by choig it u ito blocks as show i Fig blow: Fig W hav slit th ara ito rctagular blocks of width. What is th total ara of all th blocks? Ara Ara of First Block + Ara of Scod Block + Ara of Third Block +... ( ( ( What is th ara of ach block? Sic ach is a rctagular block thrfor Ara of First Block Ara of Scod Block ( Ara of Third Block ( 3 Ara + ( + ( + ( + (
4 Chatr Ifiit Sris Pag 4 of Do you rcogis th ifiit sris? It is th harmoic sris discussd arlir i this chatr. Th act ara udr th curv is giv by d whr from abov w hav d + that is it divrgs. Dos th sris covrg or divrg? By amiig Fig w ca s th aras of blocks is gratr tha th ara udr th curv giv by th itgral d + thrfor th sris +. Of cours w hav alrady rov that + bcaus it is th wll stablishd harmoic sris which divrgs. Hc th imror itgral ad th ifiit sris hav th sam covrgc. This is ot just th cas for but is grally th cas as giv by th followig roositio. f b a fuctio which is Itgral Tst (.6. Lt ( for th ( Positiv ( Cotiuous (3 Dcrasig or Costat N (grally N ad is such that f ( a ( a covrgs ( f ( d covrgs Not: Th itgral tst dos ot tll us what th sum is qual to. What dos th itgral tst say? It says that if a fuctio f satisfis th abov 3 coditios for N which mas th ( a f ( ( ( f a, f a, f 3 a,... 3 ( N ad ( a covrgs if ad oly if f ( d covrgs. This stc says that ( if w ca show ( has a fiit valu th ( a f d ( covrgs. Hc w oly d to chck th imror itgral for covrgc. Dos th itgral tst imly aythig ls? Ys it is also sayig that if ( ( a covrgs th f ( d covrgs. Rmmbr th mathmatical logic symbol mas th imlicatio gos both ways. Th cotraositiv of th statmt is that
5 Chatr Ifiit Sris Pag 5 of ( f ( d divrgs ( a divrgs Hc if th imror itgral divrgs th th ifiit sris divrgs. As i othr tsts th lowr limit may ot start at that is for K w hav K ( f ( d covrgs ( a K covrgs Proof. Omittd. W will ot rov th itgral tst bcaus w d to us som rortis of Rima itgratio which is byod our sco. F3 Alicatios of th Itgral Tst Eaml 3 Tst th followig sris for covrgc ( Solutio What fuctio, f, do w us for th itgral tst i this cas? Sic th gral trm of th giv sris is thrfor w ami th fuctio f ( whr Ca w actually aly th itgral tst? Nd to chck th 3 giv coditios of th itgral tst (.6. Sic f ( thrfor f is ositiv ad cotiuous. Also as icrass th domiator,, icrass ad so f ( dcrass. Hc all 3 coditios of th itgral tst ar satisfid. How do w us th itgral tst? Chck th covrgc of th aalogous imror itgral chck this? ( ( By valuatig th imror itgral d usig dfiitio (.5 d. But how do w ( d lim ( d (* whr is a ral ositiv umbr. Evaluatig th itgral o th Right Had Sid of (*: k d d k k ( Bcaus ( ( Substitutig Limits ad Takig Out Taki ( g i th Ngativ Sig
6 Chatr Ifiit Sris Pag 6 of Substitutig this ( ( d d lim ito (* givs lim ( Bcaus lim ( lim Th imror itgral, ( d, has a fiit valu,, thrfor it covrgs. By th itgral tst (.6 ( f ( d covrgs ( a th giv sris ( Not that covrgs. covrgs is ot th sum of th giv ifiit sris (. Th imror itgral dos ot giv th sum of th aalogous sris but it has th sam covrgc as th sris. Eaml 33 Discuss th covrgc or divrgc of l ( Solutio. W us th itgral tst for tstig th covrgc of th giv sris. Which fuctio do w us i th imror itgral? Sic th gral trm of th sris is thrfor w cosidr th fuctio l for f ( ( l (. Ar th 3 coditios of th itgral tst satisfid? f is cotiuous ad ositiv for. Sic Ys bcaus ( icrass thrfor l ( l ( icrass as dcrass so it is a dcrasig fuctio. Hc w ca us th itgral tst. How? W tst th covrgc of th aalogous imror itgral by valuatig d l (. How? Usig dfiitio (.5 d d lim l ( l ( ( How do w dtrmi th itgral o th Right Had Sid of (?
7 Chatr Ifiit Sris Pag 7 of By usig substitutio, lt u l ( du [ Diffrtiatig ] d ( du d Puttig ths ito th abov itgral givs d du / Substitutig l ( u ad d du l ( u / du l ( u l l ( Substitutig u l ( u Evaluatig th dfiit itgral d d l ( l ( Substitutig l l l ( l ( ( ( ( ( [ ] l l l l Substitutig th Limits d Puttig this rsult l ( l ( l ( l ( l ( ito ( givs d lim l ( l ( l ( l ( l ( + l l as. ( Bcaus ( d Hc th imror itgral,, dos ot hav a fiit valu so it divrgs. l ( By th cotraositiv of th itgral tst (.6 f ( d divrgs a divrgs ( ( th giv sris, l (, divrgs. Th itgral tst says that if th aalogous imror itgral divrgs th th sris divrgs. Also if th imror itgral covrgs th th sris covrgs. Eaml 34 Discuss th covrgc or divrgc of ta ( + Solutio. Ca w aly th itgral tst for th covrgc of th giv sris? Lt s try. Which fuctio do w cosidr for th imror itgral? Sic th gral trm of th sris is f ( ta ( + ta ( + w ami th fuctio ( (
8 Chatr Ifiit Sris Pag 8 of Th f is dcrasig, ositiv ad cotiuous for, thrfor all thr coditios of th itgral tst ar satisfid. Cosidr th aalogous imror itgral with > ta ( ta ( d lim d + ( + ta ( How do w fid th itgral o th Right Had Sid of (, d +? Us substitutio with u ta ( du [ Diffrtiatig ] d ( + du d + Puttig this ito th Right Had Sid itgral of ( without limits for th tim big: ta ( u Substitutig ta ( u d ( + du + + ad d ( + du u ( du Cacllig + ta ( u Substitutig u ta ta ( ta ( Substitutig this rsult d + ito th Right Had Sid of ( w hav ta ( ta ( lim d lim + ( ( ta ( ta ( lim Substitutig Limits lim ta ( ta ( π π π Bcaus lim ta ( 4 π ad ta ( 4 ta ( Th imror itgral, d, has a fiit valu, π π 4, so it covrgs. + Thrfor by th itgral tst (.6 ( th giv sris, ( f d covrgs ( a ta ( +, covrgs. covrgs
9 Chatr Ifiit Sris Pag 9 of Eaml 35 Prov th -sris tst (. of Sctio D. Th -sris is covrgs if > divrgs if Solutio. Which fuctio do w cosidr? + f :, b giv by Lt ] ] f ( whr Th f is dcrasig (or costat, ositiv ad cotiuous for ad. Hc f satisfis th coditios for usig th itgral tst. Cosidr th aalogous imror itgral d d lim ( Eami th cas for ot qual to, : What is th itgral o th Right Had Sid i ( qual to? d ( d Rwritig Cas. Lt + + >. Substitutig th abov rsult ( Itgratig ( Substitutig th limits d ito ( givs d lim Sic thrfor ( lim ( > lim ( lim ultilyig Numrator ad Domiator by Sic th imror itgral has a fiit valu thrfor it covrgs for. > By th itgral tst (.6 ( ( f d covrgs ( a covrgs th giv sris,, covrgs for. > What othr cass do w d to cosidr? Cas. Lt <. For ths valus of th imror itgral divrgs bcaus
10 Chatr Ifiit Sris Pag of d d Bca om abov lim us fr + Bcaus + as + for < Th imror itgral divrgs for < thrfor by th itgral tst (.6 ( f ( d divrgs ( a divrgs th giv sris,, divrgs for <. Cas 3. Lt. For this valu of w hav th followig imror itgral: d d lim d lim l Bcaus l ( ( lim l ( l ( ( Substitutig Limits lim l ( + Th imror itgral divrgs for thrfor by th itgral tst (.6 ( f ( d divrgs ( a th giv sris,, divrgs for. Cas 4. If < w caot us th itgral tst, why ot? divrgs Bcaus is ot a dcrasig fuctio if < so it dos ot satisfy o of th coditios for th itgral tst. W d to us aothr tst. But which o? If < th th sris divrgs bcaus th th trm dos ot covrg to. That is lim [Not Zro] for <. So by (.6 th sris divrgs for <. Collctig all th abov cass w hav rov th -sris tst: covrgs if > divrgs if Not that w ca ONLY aly th itgral tst if th 3 coditios ositiv, cotiuous ad dcrasig (or costat ar satisfid for th aalogous fuctio of th sris. ( ( (.6 If li m a th Eaml 36 a divrgs
11 Chatr Ifiit Sris Pag of Discuss th covrgc or divrgc of th followig sris (. Solutio If w wat to aly th itgral tst th w d to cosidr th aalogous fuctio of th giv gral trm. What fuctio should w cosidr? f Ca w us th itgral tst? No bcaus is a icrasig fuctio. ( ( Fig 3 Th Grah of Hc w caot aly th itgral tst bcaus it dos ot satisfy o of th coditios of th tst. So how do w tst th giv sris (? Sic th th trm divrgs. SUARY lim ( Itgral Tst (.6. Lt f ( for th [Not Zro] thrfor by (.6 th giv sris b a fuctio which is ( Positiv ( Cotiuous (3 Dcrasig or Costat N (grally N ad is such that f ( a ( d ( a f ( covrgs covrgs W iitially chck th abov thr coditios ad th tst th imror itgral for covrgc bcaus th aalogous imror itgral ad sris hav th sam covrgc. (.6 If li ( a m th ( a divrgs
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