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3 40. cos x

3 40. cos x

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Analytic Trigonometry
Section2.3: Solving Trigonometric Equations
Problem 9ECP: Solve 4tan2x+5tanx6=0.
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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.

3 40. cos x
Transcribed Image Text:3 40. cos x
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 +
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