The Integral Test. and. n 1. either both converge or both diverge. f i f 2 f 3... f n. f i f 1 f 2... f n 1. f i f x dx n 1. f i. i 1.

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1 0_090.qd //0 :5 PM Page 7 SECTIO 9. The Itegral Test ad p-series 7 Sectio 9. The Itegral Test ad p-series Use the Itegral Test to determie whether a ifiite series coverges or diverges. Use properties of p-series ad harmoic series. The Itegral Test I this ad the followig sectio, ou will stud several covergece tests that appl to series with positive terms. THEOREM 9.0 The Itegral Test If f is positive, cotiuous, ad decreasig for ad a f, the ad a f d either both coverge or both diverge. Figure 9. Iscribed rectagles: Σ i = f(i) = area a = f() a = f() a = f() a = f() Circumscribed rectagles: Σ i = f(i) = area a = f() a = f() a = f() a = f( ) Proof Begi b partitioig the iterval, ito uit itervals, as show i Figure 9.. The total areas of the iscribed rectagles ad the circumscribed rectagles are as follows. i i f i f f... f f i f f... f Iscribed area Circumscribed area The eact area uder the graph of f from to lies betwee the iscribed ad circumscribed areas. f i f d f i i i Usig the th partial sum, S f f... f, ou ca write this iequalit as S f f d S. ow, assumig that f d coverges to L, it follows that for S f L S L f. Cosequetl, S is bouded ad mootoic, ad b Theorem 9.5 it coverges. So, a coverges. For the other directio of the proof, assume that the improper itegral diverges. The approaches ifiit as ad the iequalit S f d, implies that S f d diverges. So, a diverges. OTE Remember that the covergece or divergece of a is ot affected b deletig the first terms. Similarl, if the coditios for the Itegral Test are satisfied for all >, ou ca simpl use the itegral f d to test for covergece or divergece. (This is illustrated i Eample.)

2 0_090.qd //0 :5 PM Page CHAPTER 9 Ifiite Series EXAMPLE Usig the Itegral Test Appl the Itegral Test to the series Solutio The fuctio f is positive ad cotiuous for. To determie whether f is decreasig, fid the derivative. So, f < 0 for > ad it follows that f satisfies the coditios for the Itegral Test. You ca itegrate to obtai d. So, the series diverges. b b lim f d d b b lim l. lim b lb l f() = + 5 Because the improper itegral coverges, the ifiite series also coverges. Figure 9.9 EXAMPLE Usig the Itegral Test Appl the Itegral Test to the series Solutio Because f satisfies the coditios for the Itegral Test (check this), ou ca itegrate to obtai b b d lim lim b arcta b lim arcta b arcta b. So, the series coverges (see Figure 9.9). d. TECHOLOGY I Eample, the fact that the improper itegral coverges to does ot impl that the ifiite series coverges to. To approimate the sum of the series, ou ca use the iequalit d. (See Eercise 0.) The larger the value of, the better the approimatio. For istace, usig 00 produces

3 0_090.qd //0 :5 PM Page 9 SECTIO 9. The Itegral Test ad p-series 9 HARMOIC SERIES Pthagoras ad his studets paid close attetio to the developmet of music as a abstract sciece. This led to the discover of the relatioship betwee the toe ad the legth of the vibratig strig. It was observed that the most beautiful musical harmoies correspoded to the simplest ratios of whole umbers. Later mathematicias developed this idea ito the harmoic series, where the terms i the harmoic series correspod to the odes o a vibratig strig that produce multiples of the fudametal frequec. For eample, is twice the fudametal frequec, is three times the fudametal frequec, ad so o. p-series ad Harmoic Series I the remaider of this sectio, ou will ivestigate a secod tpe of series that has a simple arithmetic test for covergece or divergece. A series of the form p p p p... p-series is a p-series, where p is a positive costat. For p, the series... Harmoic series is the harmoic series. A geeral harmoic series is of the form a b. I music, strigs of the same material, diameter, ad tesio, whose legths form a harmoic series, produce harmoic toes. The Itegral Test is coveiet for establishig the covergece or divergece of p-series. This is show i the proof of Theorem 9.. THEOREM 9. The p-series Covergece of p-series p p p p p.... coverges if p >, ad. diverges if 0 < p. Proof that The proof follows from the Itegral Test ad from Theorem.5, which states p d coverges if p > ad diverges if 0 < p. OTE The sum of the series i Eample (b) ca be show to be (This was proved b Leohard Euler, but the proof is too difficult to preset here.) Be sure ou see that the Itegral Test does ot tell ou that the sum of the series is equal to the value of the itegral. For istace, the sum of the series i Eample (b) is.5 but the value of the correspodig improper itegral is d.. EXAMPLE Coverget ad Diverget p-series Discuss the covergece or divergece of (a) the harmoic series ad (b) the p-series with p. Solutio a. From Theorem 9., it follows that the harmoic series... diverges. b. From Theorem 9., it follows that the p-series coverges.... p p

4 0_090.qd //0 :5 PM Page 0 0 CHAPTER 9 Ifiite Series EXAMPLE Testig a Series for Covergece Determie whether the followig series coverges or diverges. Solutio This series is similar to the diverget harmoic series. If its terms were larger tha those of the harmoic series, ou would epect it to diverge. However, because its terms are smaller, ou are ot sure what to epect. The fuctio f l is positive ad cotiuous for. To determie whether f is decreasig, first rewrite f as f l ad the fid its derivative. So, f < 0 for > ad it follows that f satisfies the coditios for the Itegral Test. l l f l l l l d The series diverges. l d lim b ll b lim ll b ll b OTE The ifiite series i Eample diverges ver slowl. For istace, the sum of the first 0 terms is approimatel.79, whereas the sum of the first 00 terms is just slightl larger:.507. I fact, the sum of the first 0,000 terms is approimatel you ca see that although the ifiite series adds up to ifiit, it does so ver slowl. Eercises for Sectio 9. I Eercises, use the Itegral Test to determie the covergece or divergece of the series e. e l l l l l l 5 l l l l... See for worked-out solutios to odd-umbered eercises l.... l l 5.. arcta l ll 7.. I Eercises 9 ad 0, use the Itegral Test to determie the covergece or divergece of the series, where k is a positive iteger k e k c k

5 0_090.qd //0 :5 PM Page SECTIO 9. The Itegral Test ad p-series I Eercises, eplai wh the Itegral Test does ot appl to the series.... si. I Eercises 5, use the Itegral Test to determie the covergece or divergece of the p-series I Eercises 9, use Theorem 9. to determie the covergece or divergece of the p-series I Eercises 7, match the series with the graph of its sequece of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), ad (f).] Determie the covergece or divergece of the series. (a) (c) S S 0 0 (b) (d) e cos si S S 0 0 (e) S (f) umerical ad Graphical Aalsis Use a graphig utilit to fid the idicated partial sum S ad complete the table. The use a graphig utilit to graph the first 0 terms of the sequece of partial sums. Compare the rate at which the sequece of partial sums approaches the sum of the series for each series. (a) (b) 5 5. umerical Reasoig Because the harmoic series diverges, it follows that for a positive real umber M there eists a positive iteger such that the partial sum > M S (a) Use a graphig utilit to complete the table. M (b) As the real umber M icreases i equal icremets, does the umber icrease i equal icremets? Eplai. 5 Writig About Cocepts 5. State the Itegral Test ad give a eample of its use.. Defie a p-series ad state the requiremets for its covergece. 7. A fried i our calculus class tells ou that the followig series coverges because the terms are ver small ad approach 0 rapidl. Is our fried correct? Eplai. 0,000 0,00 0,00... S 0

6 0_090.qd //0 :5 PM Page CHAPTER 9 Ifiite Series Writig About Cocepts (cotiued). I Eercises 7, lim a 0 for each series but the do ot all coverge. Is this a cotradictio of Theorem 9.9? Wh do ou thik some coverge ad others diverge? Eplai. 9. Use a graph to show that > d. What ca ou coclude about the covergece or divergece of the series? Eplai. 50. Let f be a positive, cotiuous, ad decreasig fuctio for, such that a f. Use a graph to rak the followig quatities i decreasig order. Eplai our reasoig. (a) (b) f d (c) I Eercises 5 5, fid the positive values of p for which the series coverges. l l p p p p I Eercises 55 5, use the result of Eercise 5 to determie the covergece or divergece of the series l l l l 59. Let f be a positive, cotiuous, ad decreasig fuctio for, such that a f. Prove that if the series a coverges to S, the the remaider R S S is bouded b 0 R f d. 7 a f 7 R + R + R + a 0. Show that the result of Eercise 59 ca be writte as f d. a a a I Eercises, use the result of Eercise 59 to approimate the sum of the coverget series usig the idicated umber of terms. Iclude a estimate of the maimum error for our approimatio.. si terms,. four terms 5,. te terms,. te terms l, 5. e, four terms. e, four terms I Eercises 7 7, use the result of Eercise 59 to fid such that R 0.00 for the coverget series e e (a) Show that coverges ad diverges.. l (b) Compare the first five terms of each series i part (a). (c) Fid > such that. < l. 7. Te terms are used to approimate a coverget p-series. Therefore, the remaider is a fuctio of p ad is 0 R 0 p d, p 0 p >. (a) Perform the itegratio i the iequalit. (b) Use a graphig utilit to represet the iequalit graphicall. (c) Idetif a asmptotes of the error fuctio ad iterpret their meaig. +

7 0_090.qd //0 :5 PM Page SECTIO 9. The Itegral Test ad p-series 75. Euler s Costat Let S k (a) Show that l S l. (b) Show that the sequece a S l is bouded. (c) Show that the sequece a is decreasig. (d) Show that a coverges to a limit (called Euler s costat). (e) Approimate usig a Fid the sum of the series l. 77. Cosider the series l. k.... (a) Determie the covergece or divergece of the series for. (b) Determie the covergece or divergece of the series for e. (c) Fid the positive values of for which the series coverges. 7. The Riema zeta fuctio for real umbers is defied for all for which the series coverges. Fid the domai of the fuctio. Review I Eercises 79 90, determie the covergece or divergece of the series l l l Sectio Project: The Harmoic Series The harmoic series is oe of the most importat series i this chapter. Eve though its terms ted to zero as icreases, lim 0 the harmoic series diverges. I other words, eve though the terms are gettig smaller ad smaller, the sum adds up to ifiit. (a) Oe wa to show that the harmoic series diverges is attributed to Jakob Beroulli. He grouped the terms of the harmoic series as follows: > > > (b) Use the proof of the Itegral Test, Theorem 9.0, to show that l... l. (c) Use part (b) to determie how ma terms M ou would eed so that M > 50. (d) Show that the sum of the first millio terms of the harmoic series is less tha 5. (e) Show that the followig iequalities are valid. l l 0 9 l l 99 (f) Use the ideas i part (e) to fid the limit lim m m. m > Write a short paragraph eplaiig how ou ca use this groupig to show that the harmoic series diverges.