11.2. Infinite Series

Transcription

1 .2 Infinite Series 76.2 Infinite Series An infinite series is the sum f n infinite seuence f numbers Á + n + Á The gl f this sectin is t understnd the mening f such n infinite sum nd t develp methds t clculte it. Since there re infinitely mny terms t dd in n infinite series, we cnnt just keep dding t see wht cmes ut. Insted we lk t wht we get by summing the first n terms f the seuence nd stpping. The sum f the first n terms s n = Á + n

2 762 Chpter : Infinite Seuences nd Series is n rdinry finite sum nd cn be clculted by nrml dditin. It is clled the nth prtil sum. As n gets lrger, we expect the prtil sums t get clser nd clser t limiting vlue in the sme sense tht the terms f seuence pprch limit, s discussed in Sectin.. Fr exmple, t ssign mening t n expressin like Á We dd the terms ne t time frm the beginning nd lk fr pttern in hw these prtil sums grw. Suggestive expressin fr Prtil sum prtil sum Vlue First: s = 2-3 Secnd: s 2-2 = Third: s 2-3 = nth: s Á + 2 n n - 2 n - 2 n - Indeed there is pttern. The prtil sums frm seuence whse nth term is s n = 2-2 n -. This seuence f prtil sums cnverges t 2 becuse lim n: s>2 n d = 0. We sy the sum f the infinite series Á + 2 n - + Á is 2. Is the sum f ny finite number f terms in this series eul t 2? N. Cn we ctully dd n infinite number f terms ne by ne? N. But we cn still define their sum by defining it t be the limit f the seuence f prtil sums s n :, in this cse 2 (Figure.5). Our knwledge f seuences nd limits enbles us t brek wy frm the cnfines f finite sums. /4 0 /2 /8 2 FIGURE.5 As the lengths, 2, 4, 8, Á re dded ne by ne, the sum pprches 2.

3 .2 Infinite Series 763 HISTORICAL BIOGRAPHY Blise Pscl ( ) DEFINITIONS Infinite Series, nth Term, Prtil Sum, Cnverges, Sum Given seuence f numbers 5 n 6, n expressin f the frm Á + n + Á is n infinite series. The number 5s n 6 defined by n is the nth term f the series. The seuence s = s 2 = + 2 s n = Á + n = n is the seuence f prtil sums f the series, the number s n being the nth prtil sum. If the seuence f prtil sums cnverges t limit L, we sy tht the series cnverges nd tht its sum is L. In this cse, we ls write Á + n + Á = If the seuence f prtil sums f the series des nt cnverge, we sy tht the series diverges. k k = n = L. When we begin t study given series Á + n + Á, we might nt knw whether it cnverges r diverges. In either cse, it is cnvenient t use sigm nttin t write the series s Gemetric Series n, k, r n k = Gemetric series re series f the frm + r + r 2 + Á + r n - + Á = A useful shrthnd when summtin frm t is understd r n - in which nd r re fixed rel numbers nd Z 0. The series cn ls be written s g n=0 r n. The rti r cn be psitive, s in Á + 2 b n - + Á, r negtive, s in Á b n - + Á. If r =, the nth prtil sum f the gemetric series is s n = + sd + sd 2 + Á + sd n - = n,

4 764 Chpter : Infinite Seuences nd Series nd the series diverges becuse lim n: sn = ;, depending n the sign f. If r = -, the series diverges becuse the nth prtil sums lternte between nd 0. If ƒ r ƒ Z, we cn determine the cnvergence r divergence f the series in the fllwing wy: s n = + r + r 2 + Á + r n - rs n = r + r 2 + Á + r n - + r n s n - rs n = - r n s n s - rd = s - r n d s n = s - r n d, sr Z d. - r Multiply s n by r. Subtrct rs n frm s n. Mst f the terms n the right cncel. Fctr. We cn slve fr s n if r Z. If ƒ r ƒ 6, then r n : 0 s n : (s in Sectin.) nd s n : >s - rd. If ƒ r ƒ 7, then ƒ r n ƒ : nd the series diverges. If ƒ r ƒ 6, the gemetric series + r + r 2 + Á + r n - + Á cnverges t >s - rd: r n - = If ƒ r ƒ Ú, the series diverges. - r, ƒ r ƒ 6. We hve determined when gemetric series cnverges r diverges, nd t wht vlue. Often we cn determine tht series cnverges withut knwing the vlue t which it cnverges, s we will see in the next severl sectins. The frmul >s - rd fr the sum f gemetric series pplies nly when the summtin index begins with in the expressin g (r with the index n = 0 if we write the series s g n= r n - n=0 r n ). EXAMPLE Index Strts with The gemetric series with = >9 nd r = >3 is EXAMPLE 2 The series Index Strts with n = 0 is gemetric series with = 5 nd r = ->4. It cnverges t EXAMPLE Á = n = 0 A Buncing Bll n b s -d n 5 4 n = Á - r = 5 + s>4d = 4. Yu drp bll frm meters bve flt surfce. Ech time the bll hits the surfce fter flling distnce h, it rebunds distnce rh, where r is psitive but less thn. Find the ttl distnce the bll trvels up nd dwn (Figure.6). = >9 - s>3d = 6.

5 .2 Infinite Series 765 Slutin The ttl distnce is r r 2 r 3 s = + 2r + 2r 2 + 2r 3 + Á = + (''''''')'''''''* This sum is 2r>s - rd. If = 6 m nd r = 2>3, fr instnce, the distnce is s = 6 + s2>3d - s2>3d = 6 5>3 b = 30 m. >3 2r - r = + r - r. EXAMPLE 4 Repeting Decimls Express the repeting deciml Á s the rti f tw integers. () Slutin Á = s00d s00d 3 + Á = b + Á b ('''''''')''''''''* >s - 0.0d = b = = =, r = >00 Unfrtuntely, frmuls like the ne fr the sum f cnvergent gemetric series re rre nd we usully hve t settle fr n estimte f series sum (mre but this lter). The next exmple, hwever, is nther cse in which we cn find the sum exctly. EXAMPLE 5 Find the sum f the series A Nngemetric but Telescping Series nsn + d. (b) FIGURE.6 () Exmple 3 shws hw t use gemetric series t clculte the ttl verticl distnce trveled by buncing bll if the height f ech rebund is reduced by the fctr r. (b) A strbscpic pht f buncing bll. Slutin We lk fr pttern in the seuence f prtil sums tht might led t frmul fr s k. The key bservtin is the prtil frctin decmpsitin s nd k nsn + d = nsn + d = n - n +, k n - n + b s k = - 2 b b b + Á + k - k + b. Remving prentheses nd cnceling djcent terms f ppsite sign cllpses the sum t s k = - k +.

6 766 Chpter : Infinite Seuences nd Series We nw see tht s k : s k :. The series cnverges, nd its sum is : nsn + d =. Divergent Series One resn tht series my fil t cnverge is tht its terms dn t becme smll. EXAMPLE 6 () The series Prtil Sums Outgrw Any Number n 2 = Á + n 2 + Á diverges becuse the prtil sums grw beynd every number L. After the prtil sum s is greter thn n Á, + n 2. (b) The series n + n = Á + n + n + Á diverges becuse the prtil sums eventully utgrw every pressigned number. Ech term is greter thn, s the sum f n terms is greter thn n. The nth-term Test fr Divergence Observe tht lim must eul zer if the series cnverges. T see why, let S represent the series sum nd s n = Á g n: n n= n + n the nth prtil sum. When n is lrge, bth s n nd s n - re clse t S, s their difference, n, is clse t zer. Mre frmlly, Difference Rule fr n = s n - s n - : S - S = 0. seuences This estblishes the fllwing therem. Cutin Therem 7 des nt sy tht g n= n cnverges if n : 0. It is pssible fr series t diverge when n : 0. THEOREM 7 If n cnverges, then n : 0. Therem 7 leds t test fr detecting the kind f divergence tht ccurred in Exmple 6. The nth-term Test fr Divergence n diverges if lim n: n fils t exist r is different frm zer.

7 .2 Infinite Series 767 EXAMPLE 7 () (b) Applying the nth-term Test diverges becuse n 2 : diverges becuse (c) s -d n + diverges becuse lim n: s -d n + des nt exist (d) n 2 n + n -n 2n + 5 diverges becuse n + n : lim n: -n 2n + 5 =- 2 Z 0. EXAMPLE 8 The series n : 0 but the Series Diverges Á + 2 n + 2 n + Á + 2 n + Á (')'* ('''')''''* 2 terms 4 terms (''''')'''''* 2 n terms diverges becuse the terms re gruped int clusters tht dd t, s the prtil sums increse withut bund. Hwever, the terms f the series frm seuence tht cnverges t 0. Exmple f Sectin.3 shws tht the hrmnic series ls behves in this mnner. Cmbining Series Whenever we hve tw cnvergent series, we cn dd them term by term, subtrct them term by term, r multiply them by cnstnts t mke new cnvergent series. THEOREM 8 If g n = A nd gb n = B re cnvergent series, then. Sum Rule: gs n + b n d = g n + gb n = A + B 2. Difference Rule: gs n - b n d = g n - gb n = A - B 3. Cnstnt Multiple Rule: gk n = kg n = ka sany number kd. Prf The three rules fr series fllw frm the nlgus rules fr seuences in Therem, Sectin.. T prve the Sum Rule fr series, let A n = Á + n, B n = b + b 2 + Á + b n. Then the prtil sums f gs n + b n d re s n = s + b d + s 2 + b 2 d + Á + s n + b n d = s + Á + n d + sb + Á + b n d = A n + B n.

8 768 Chpter : Infinite Seuences nd Series Since A n : A nd B n : B, we hve s n : A + B by the Sum Rule fr seuences. The prf f the Difference Rule is similr. T prve the Cnstnt Multiple Rule fr series, bserve tht the prtil sums f gk n frm the seuence s n = k + k 2 + Á + k n = ks Á + n d = ka n, which cnverges t ka by the Cnstnt Multiple Rule fr seuences. As crllries f Therem 8, we hve. Every nnzer cnstnt multiple f divergent series diverges. 2. If g n cnverges nd gb n diverges, then gs n + b n d nd gs n - b n d bth diverge. We mit the prfs. CAUTION Remember tht gs n + cn cnverge when nd bth diverge. Fr exmple, g nd diverge, wheres gs n + b n d = Á gb n = s -d + s -d + s -d + Á Á b n d g n gb n cnverges t 0. EXAMPLE 9 Find the sums f the fllwing series. () 3 n n - = = = 2 n n - b 2 n n - - s>2d - - s>6d Difference Rule Gemetric series with = nd r = >2, >6 = = 4 5 (b) n = n = 4 n = 0 2 n = 4 - s>2d b = 8 Cnstnt Multiple Rule Gemetric series with =, r = >2 Adding r Deleting Terms We cn dd finite number f terms t series r delete finite number f terms withut ltering the series cnvergence r divergence, lthugh in the cse f cnvergence this will usully chnge the sum. If g cnverges, then g n= n n=k n cnverges fr ny k 7 nd n = Á + k - + n. Cnversely, if g cnverges fr ny then g n=k n k 7, n= n cnverges. Thus, n = 4 n = k 5 n

9 .2 Infinite Series 769 nd n = 4 5 n = 5 n b HISTORICAL BIOGRAPHY Richrd Dedekind (83 96) Reindexing As lng s we preserve the rder f its terms, we cn reindex ny series withut ltering its cnvergence. T rise the strting vlue f the index h units, replce the n in the frmul fr by n - h: n n = T lwer the strting vlue f the index h units, replce the n in the frmul fr n + h: n = n - h = Á. + h n + h = Á. - h It wrks like hrizntl shift. We sw this in strting gemetric series with the index n = 0 insted f the index, but we cn use ny ther strting index vlue s well. We usully give preference t indexings tht led t simple expressins. n by EXAMPLE 0 We cn write the gemetric series s Reindexing Gemetric Series n = 0 2 n, n = 5 2 n - = Á 2 n - 5, r even The prtil sums remin the sme n mtter wht indexing we chse. n =-4 2 n + 4.