To determine To find: The Taylor series for the function f ( x ) = ln ( 1 − x 2 ) and also find its interval of convergence. Explanation Given that, the function is f ( x ) = ln ( 1 + 4 x ) . Note that, Taylor series for the function ln ( 1 + x ) is x − x 2 2 + x 3 3 − x 4 4 + ⋯ + ( − 1 ) n x n + 1 n + 1 + ⋯ and the interval of convergence is ( − 1 , 1 ] . Assume that a function f has a Taylor series such that f ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ + a n x n + ⋯ . Then replace each x with g ( x ) = c x k for some constant c and positive integer k , which on solving gives the Taylor series for f [ g ( x ) ] as, a 0 + a 1 g ( x ) + a 2 [ g ( x ) ] 2 + a 3 [ g ( x ) ] 3 + ⋯ + a n [ g ( x ) ] n + ⋯ . In this case the interval of convergence of new series can vary. Obtain the Taylor series for the the given function can be replacing x with ( − x 2 ) in the Taylor series of ln ( 1 + x ) . Thus, the Taylor series of given function is computed as follows

Calculus with Applications Books a la Carte Edition

11th Edition

ISBN: 9780133864564

Author: Margaret L. Lial; Nathan P. Ritchey; Raymond N. Greenwell

Publisher: Pearson Education

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Chapter 12.5, Problem 12E

To determine

To find: The Taylor series for the function f(x)=ln(1−x2) and also find its interval of convergence.

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Chapter 12.5, Problem 11E

Chapter 12.5, Problem 13E

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Calculus with Applications Books a la Carte Edition

Ch. 12.1 - Find the first four terms of the sequence having...Ch. 12.1 - Prob. 2YT Ch. 12.1 - Prob. 3YT Ch. 12.1 - Prob. 4YT Ch. 12.1 - Prob. 5YT Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - List the first n terms of the geometric sequence...Ch. 12.1 - Prob. 5E

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