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Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1

Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 49E
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Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

x tan-1 x2

Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1 <xs1 k=1 x x3 x* -In (1 - x) = x + for -1 sx <1 ... %3D k=1 (-1)*x*+1 Σ (-1)* x*+1 tanx = x for |x| s 1 2k + 1 2k + 1 x*+1 x2k+1 Σ for |x| < 0 sinh x = x + 5! (2k + 1)! (2k + 1)!' x* cosh x for |x| < 0 +.. 2! 4! (2k)! (2k)!" moted in Theorem 11.6 the hinoial erieay converge to (1 at +1 denendine on the value of a (3) p(p – 1)(p – 2) - (p – k + 1) (1 + x)P = E()x 2(3) - r*, for x| < 1 and k!
Transcribed Image Text:Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1 <xs1 k=1 x x3 x* -In (1 - x) = x + for -1 sx <1 ... %3D k=1 (-1)*x*+1 Σ (-1)* x*+1 tanx = x for |x| s 1 2k + 1 2k + 1 x*+1 x2k+1 Σ for |x| < 0 sinh x = x + 5! (2k + 1)! (2k + 1)!' x* cosh x for |x| < 0 +.. 2! 4! (2k)! (2k)!" moted in Theorem 11.6 the hinoial erieay converge to (1 at +1 denendine on the value of a (3) p(p – 1)(p – 2) - (p – k + 1) (1 + x)P = E()x 2(3) - r*, for x| < 1 and k!
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