Explanation Result used: (1) If f has a power series expansion at a , f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n , f ( x ) = f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ‴ ( a ) 3 ! ( x − a ) 3 + ⋅ ⋅ ⋅ (2) The Ratio Test : (i) If lim n → ∞ | a n + 1 a n | = L < 1 , then the series ∑ n = 1 ∞ a n is absolutely convergent (and therefore convergent.) (ii) If lim n → ∞ | a n + 1 a n | = L > 1 or lim n → ∞ | a n + 1 a n | = ∞ , then the series ∑ n = 1 ∞ a n is divergent. (ii) If lim n → ∞ | a n + 1 a n | = 1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of ∑ n = 1 ∞ a n . Calculation: Consider the function f ( x ) = cos x centered at a = π 2 . Since the function f ( x ) = cos x at π 2 is f ( π 2 ) = 0 . Find the first derivative of f ( x ) at a = π 2 . f ′ ( x ) = d d x ( cos x ) = − sin x f ′ ( x ) = − sin x (1) Obtain f ′ ( π 2 ) . f ′ ( π 2 ) = − sin ( π 2 ) = − 1 Find the second derivative of f ( x ) at a = π 2 . f ( 2 ) ( x ) = d 2 d x 2 ( cos x ) = d d x ( f ′ ( x ) ) [ ∵ d d x ( cos x ) = f ′ ( x ) ] = d d x ( − sin x ) ( by equation(1) ) = − cos x That is, f ( 2 ) ( x ) = − cos x (2) Compute f ( 2 ) ( π 2 ) . f ( 2 ) ( π 2 ) = − cos ( π 2 ) = 0 That is, f ( 2 ) ( π 2 ) = 0 Find the third derivative of f ( x ) at a = π 2 as follows. f ( 3 ) ( x ) = d 3 d x 3 ( cos x ) = d d x ( f ( 2 ) ( x ) ) = d d x ( − cos x ) ( by equation(2) ) = − ( − sin x ) That is, f ( 3 ) ( x ) = sin x (3) Compute f ( 3 ) ( π 2 ) . f ( 3 ) ( π 2 ) = sin ( π 2 ) = 1 Find the fourth derivative of f ( x ) at a = π 2 . f ( 4 ) ( x ) = d 4 d x 4 ( cos x ) = d d x ( f ( 3 ) ( x ) ) = d d x ( sin x ) ( by equation(3) ) = cos x That is, f ( 4 ) ( x ) = cos x (4) Compute f ( 4 ) ( π 2 ) . f ( 4 ) ( π 2 ) = cos ( π 2 ) = 0 Find the fifth derivative of f ( x ) at a = π 2 . f ( 5 ) ( x ) = d 5 d x 5 ( cos x ) = d d x ( f ( 3 ) ( x ) ) = d d x ( cos x ) ( by equation(3) ) = − sin x That is, f ( 5 ) ( x ) = − sin x (4) Compute f ( 5 ) ( π 2 )

The Taylor series for f ( x ) centered at π 2 and radius of convergence.

Question

Chapter 11.10, Problem 24E

To determine

To find: The Taylor series for f(x) centered at π2 and radius of convergence.

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Chapter 11.10, Problem 23E

Chapter 11.10, Problem 25E

Multivariable Calculus

8th Edition

ISBN: 9781305266643

Author: James Stewart

Publisher: Cengage Learning

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Multivariable Calculus

Infinite Sequences And Series. 24E

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Chapter 11 Solutions

Multivariable Calculus

Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E

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