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36. sin x?

36. sin 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.

36. sin x?
Transcribed Image Text:36. sin 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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