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EXAMPLE 9 Find the Maclaurin series for the function below and its radius of convergence. f (z) = V16

EXAMPLE 9 Find the Maclaurin series for the function below and its radius of convergence. f (z) = V16

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Sequences And Series
Section8.3: Geometric Sequences
Problem 73E
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Video Example ) EXAMPLE 9 Find the Maclaurin series for the function below and its radius of convergence. f (x) = V16 – x SOLUTION We write f(x) in a form where we can use the binomial series. V16 - x (1 - x/16) 5 =-(1 - x/16)-1/2 4 Using the binomial series with k = and with x replaced by -x/16, we have 5 5 00 (1 - )-1/2 Σ V16 - x n 4 n=0 5 (-1/2)(-3/2)( = -[1 + (-1/2)(-–) +. (--)2 + 4 16 2! 16 3! 16 (-1/2)(-3/2)(-5/2)...(-1/2 - n + 1) + ... + (--)" + ...] 16 n! 1 1: 3 1:3· 5 1· 3· 5... · (2n - 1) =-[1 + + X + + ... + x² + ...] 4 2!322 3!323 n!32" We know from the binomial series that this series converges when | -x/16 | < 1, that is | × | < , so the radius of convergence is R = Need Help? Read It
Transcribed Image Text:Video Example ) EXAMPLE 9 Find the Maclaurin series for the function below and its radius of convergence. f (x) = V16 – x SOLUTION We write f(x) in a form where we can use the binomial series. V16 - x (1 - x/16) 5 =-(1 - x/16)-1/2 4 Using the binomial series with k = and with x replaced by -x/16, we have 5 5 00 (1 - )-1/2 Σ V16 - x n 4 n=0 5 (-1/2)(-3/2)( = -[1 + (-1/2)(-–) +. (--)2 + 4 16 2! 16 3! 16 (-1/2)(-3/2)(-5/2)...(-1/2 - n + 1) + ... + (--)" + ...] 16 n! 1 1: 3 1:3· 5 1· 3· 5... · (2n - 1) =-[1 + + X + + ... + x² + ...] 4 2!322 3!323 n!32" We know from the binomial series that this series converges when | -x/16 | < 1, that is | × | < , so the radius of convergence is R = Need Help? Read It
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