Representation of Functions by Power Series. Geometric Power Series

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1 60_0909.qd //0 :09 PM Page 669 SECTION 9.9 Representation of Functions b Power Series 669 The Granger Collection Section 9.9 JOSEPH FOURIER (768 80) Some of the earl work in representing functions b power series was done b the French mathematician Joseph Fourier. Fourier s work is important in the histor of calculus, partl because it forced eighteenth centur mathematicians to question the then-prevailing narrow concept of a function. Both Cauch and Dirichlet were motivated b Fourier s work with series, and in 87 Dirichlet published the general definition of a function that is used toda. Representation of Functions b Power Series Find a geometric power series that represents a function. Construct a power series using series operations. Geometric Power Series In this section and the net, ou will stud several techniques for finding a power series that represents a given function. Consider the function given b f. The form of f closel resembles the sum of a geometric series ar n a n0 r, In other words, if ou let a and r, a power series representation for, centered at 0, is n n0 Of course, this series represents onl on the interval,, whereas f is defined for all, as shown in Figure 9.. To represent f in another interval, ou must develop a different series. For instance, to obtain the power series centered at, ou could write a r which implies that a and r. So, for ou have r <...., n0 which converges on the interval,. <. f n <, 8..., < f() =, Domain: all f() = Σ n, Domain: < < n = 0 Figure 9.

2 60_0909.qd //0 :09 PM Page CHAPTER 9 Infinite Series EXAMPLE Finding a Geometric Power Series Centered at 0 Find a power series for f centered at 0., Long Division... ) Solution Writing f in the form a r produces a r which implies that a and r. So, the power series for f is This power series converges when < n0 n0 ar n n which implies that the interval of convergence is,. Another wa to determine a power series for a rational function such as the one in Eample is to use long division. For instance, b dividing into, ou obtain the result shown at the left. EXAMPLE Finding a Geometric Power Series Centered at Find a power series for f centered at., Solution Writing f in the form a r produces a r which implies that a and r. So, the power series for f is ar n n0 n n0 n n n0.... This power series converges when < which implies that the interval of convergence is 0,.

3 60_0909.qd //0 :09 PM Page 67 SECTION 9.9 Representation of Functions b Power Series 67 Operations with Power Series The versatilit of geometric power series will be shown later in this section, following a discussion of power series operations. These operations, used with differentiation and integration, provide a means of developing power series for a variet of elementar functions. (For simplicit, the following properties are stated for a series centered at 0.) Operations with Power Series Let f a and g b n n n n.. fk a n k n n n0. f N a n nn n0. f ± g a n ± b n n n0 The operations described above can change the interval of convergence for the resulting series. For eample, in the following addition, the interval of convergence for the sum is the intersection of the intervals of convergence of the two original series. n n0 n0 n n0 nn,,, EXAMPLE Adding Two Power Series Find a power series, centered at 0, for f. Solution Using partial fractions, ou can write f as B adding the two geometric power series and. ou obtain the following power series. n0 n0 n0 n, n n, < < n n... The interval of convergence for this power series is,.

4 60_0909.qd //0 :09 PM Page CHAPTER 9 Infinite Series EXAMPLE Finding a Power Series b Integration Find a power series for f ln, centered at. Solution From Eample, ou know that n n. n0 Integrating this series produces Interval of convergence: 0, ln d C C B letting, ou can conclude that C 0. Therefore, ln n0 n0 n n. n n n n Note that the series converges at. This is consistent with the observation in the preceding section that integration of a power series ma alter the convergence at the endpoints of the interval of convergence..... Interval of convergence: 0, TECHNOLOGY In Section 9.7, the fourth-degree Talor polnomial for the natural logarithmic function ln was used to approimate ln.. ln You now know from Eample that this polnomial represents the first four terms of the power series for ln. Moreover, using the Alternating Series Remainder, ou can determine that the error in this approimation is less than R a During the seventeenth and eighteenth centuries, mathematical tables for logarithms and values of other transcendental functions were computed in this manner. Such numerical techniques are far from outdated, because it is precisel b such means that man modern calculating devices are programmed to evaluate transcendental functions.

5 60_0909.qd //0 :09 PM Page 67 SECTION 9.9 Representation of Functions b Power Series 67 EXAMPLE 5 Finding a Power Series b Integration Find a power series for g arctan, centered at 0. The Granger Collection SRINIVASA RAMANUJAN (887 90) Series that can be used to approimate have interested mathematicians for the past 00 ears. An amazing series for approimating was discovered b the Indian mathematician Srinivasa Ramanujan in 9 (see Eercise 6). Each successive term of Ramanujan s series adds roughl eight more correct digits to the value of. For more information about Ramanujan s work, see the article Ramanujan and Pi b Jonathan M. Borwein and Peter B. Borwein in Scientific American. Solution Because D arctan, ou can use the series f Substituting for produces Finall, b integrating, ou obtain arctan d C n0 n0 f C n0 n0 n n. n n n n n n 5 5 n n Interval of convergence:, Let 0, then C 0. Interval of convergence:, It can be shown that the power series developed for arctan in Eample 5 also converges (to arctan ) for ±. For instance, when, ou can write arctan However, this series (developed b James Gregor in 67) does not give us a practical wa of approimating because it converges so slowl that hundreds of terms would have to be used to obtain reasonable accurac. Eample 6 shows how to use two different arctangent series to obtain a ver good approimation of using onl a few terms. This approimation was developed b John Machin in 706. EXAMPLE 6 Approimating with a Series Use the trigonometric identit arctan 5 arctan 9 to approimate the number [see Eercise 50(b)]. Solution B using onl five terms from each of the series for arctan5 and arctan9, ou obtain arctan 5 arctan which agrees with the eact value of with an error of less than

6 60_0909.qd //0 :09 PM Page CHAPTER 9 Infinite Series In Eercises, find a geometric power series for the function, centered at 0, (a) b the technique shown in Eamples and and (b) b long division.. f.. f. In Eercises 5 6, find a power series for the function, centered at c, and determine the interval of convergence. 5. f c 5 6., 7. f c 0 8., 9. g c 0. 5,. f c 0., In Eercises 7 6, use the power series to determine a power series, centered at 0, for the function. Identif the interval of convergence Eercises for Section 9.9 g g, 7, f, f, n0 n n c 0 c 0 c 0 c 0 7. h 8. h 9. f d d f 5 f f 5, f, h 5, f, f d f ln d d f ln d d. g. f ln 5. h 6. f arctan c c c 0 c Graphical and Numerical Analsis S n... In Eercises 7 and 8, let Use a graphing utilit to confirm the inequalit graphicall. Then complete the table to confirm the inequalit numericall. S n 7. S ln S 8. S ln S 5 In Eercises 9 and 0, (a) graph several partial sums of the series, (b) find the sum of the series and its radius of convergence, (c) use 50 terms of the series to approimate the sum when 0.5, and (d) determine what the approimation represents and how good the approimation is In Eercises, match the polnomial approimation of the function f arctan with the correct graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) See for worked-out solutions to odd-numbered eercises. ln S n n n n n n n n0 n! (b) (d). g. n n g. g 5. g

7 60_0909.qd //0 :09 PM Page 675 SECTION 9.9 Representation of Functions b Power Series 675 In Eercises 5 8, use the series for f arctan to approimate the value, using R N arctan 6. arctan 7. d 8. 0 In Eercises 9, use the power series Find the series representation of the function and determine its interval of convergence.. Probabilit A fair coin is tossed repeatedl. The probabilit that the first head occurs on the nth toss is Pn n. When this game is repeated man times, the average number of tosses required until the first head occurs is (This value is called the epected value of n. ) Use the results of Eercises 9 to find En. Is the answer what ou epected? Wh or wh not?. Use the results of Eercises 9 to find the sum of each series. (a) Writing series g (b) In Eercises 5 8, eplain how to use the geometric to find the series for the function. Do not find the series. 5. f f 5 8. f ln 9. Prove that arctan arctan arctan for provided the value of the left side of the equation is between and. 50. Use the result of Eercise 9 to verif each identit. (a) (b) n0 En n, n npn. n n n <. 9. f 0. f. f. f n0 n, < arctan 0 9 arctan 9 arctan 5 arctan 9 0 n n 9 0 n [Hint: Use Eercise 9 twice to find arctan 5. Then use part (a).] 0 0 arctan d arctan d f In Eercises 5 and 5, (a) verif the given equation and (b) use the equation and the series for the arctangent to approimate to two-decimal-place accurac. 5. arctan arctan 7 5. arctan arctan In Eercises 5 58, find the sum of the convergent series b using a well-known function. Identif the function and eplain how ou obtained the sum. 5. n 5. n n n 55. n n n n n 57. n 58. n n n0 Writing About Concepts 6. Use a graphing utilit to show that n!0 6,90n. n0 n!96 n (Note: This series was discovered b the Indian mathematician Srinivasa Ramanujan in 9.) In Eercises 65 and 66, find the sum of the series. 65. n 66. n n n0 n n n n n n0 n n n0 n n n 59. Use the results of Eercises to make a geometric argument for wh the series approimations of f arctan have onl odd powers of. 60. Use the results of Eercises to make a conjecture about the degrees of series approimations of f arctan that have relative etrema. 6. One of the series in Eercises 5 58 converges to its sum at a much lower rate than the other five series. Which is it? Eplain wh this series converges so slowl. Use a graphing utilit to illustrate the rate of convergence. 6. The radius of convergence of the power series a n n n0 is. What is the radius of convergence of the series na n n? Eplain. n 6. The power series a converges for <. n n What can ou conclude about the series Eplain. n0 a n n n0 n? n n n n!