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cosh (-2x)

cosh (-2x)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.

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 Table 11.5 1 = 1 + x + x² + 1- x + x* + Ert, for |x| < 1 ... - k=0 convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f centered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). 1 = 1 - x + x² . - E(-1)x*, for [x| < 1 - ..·+ (-1)*x* + • · · = | 1 + x k=0 for |x| < 0 k! e = 1 + x + + + + ... - 2! k! k=0 (-1)*x*+1 (2k + 1)! (-1)*x*+1 (2k + 1)! sin x = x – + for x < 0 +... = If the series converges at x = -R, then it 3! 5! k=0 converges to lim f(x). x--R+ (-1)*x* Σ (2k)! (-1)* x* 2k x2 + 2! 00 For example, this theorem would for |x| < ∞ cos x = 1 +...= allow us to conclude that the series for 4! (2k)! k=0 In (1 + x) converges to In 2 at x = 1. x? In (1 + x) = x - (-1)*+1 * + (-1)*+'* Σ for -1 < x< 1 +... = 3 k k=1 -In (1 – x) : x? = x + + + k for -1 < x < 1 3 k=1 (-1)*x*+1 (-1)* x*+1 -1 tan Σ 2k + 1 for |x| < 1 n¯'x = x - + +...= 3 2k + 1 k=0 x2k+1 Σ k=o(2k + 1)!’ x2k+1 sinh x = x + 3! for x < 0 +...= 5! (2k + 1)! cosh x = 1 + + 4! Σ for x < 0 +... = (2k)! (2k)! k=0 p(p – 1)(p – 2) · ·· (p – k + 1) (P) : As noted in Theorem 11.6, the binomial series may converge to (1 + x)® at x = ±1, depending on the value of p. (2) ... (1 + x)" = Σ x*, for x < 1 and = 1 k! =0 +
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 Table 11.5 1 = 1 + x + x² + 1- x + x* + Ert, for |x| < 1 ... - k=0 convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f centered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). 1 = 1 - x + x² . - E(-1)x*, for [x| < 1 - ..·+ (-1)*x* + • · · = | 1 + x k=0 for |x| < 0 k! e = 1 + x + + + + ... - 2! k! k=0 (-1)*x*+1 (2k + 1)! (-1)*x*+1 (2k + 1)! sin x = x – + for x < 0 +... = If the series converges at x = -R, then it 3! 5! k=0 converges to lim f(x). x--R+ (-1)*x* Σ (2k)! (-1)* x* 2k x2 + 2! 00 For example, this theorem would for |x| < ∞ cos x = 1 +...= allow us to conclude that the series for 4! (2k)! k=0 In (1 + x) converges to In 2 at x = 1. x? In (1 + x) = x - (-1)*+1 * + (-1)*+'* Σ for -1 < x< 1 +... = 3 k k=1 -In (1 – x) : x? = x + + + k for -1 < x < 1 3 k=1 (-1)*x*+1 (-1)* x*+1 -1 tan Σ 2k + 1 for |x| < 1 n¯'x = x - + +...= 3 2k + 1 k=0 x2k+1 Σ k=o(2k + 1)!’ x2k+1 sinh x = x + 3! for x < 0 +...= 5! (2k + 1)! cosh x = 1 + + 4! Σ for x < 0 +... = (2k)! (2k)! k=0 p(p – 1)(p – 2) · ·· (p – k + 1) (P) : As noted in Theorem 11.6, the binomial series may converge to (1 + x)® at x = ±1, depending on the value of p. (2) ... (1 + x)" = Σ x*, for x < 1 and = 1 k! =0 +
44. cosh (-2x)
Transcribed Image Text:44. cosh (-2x)
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