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Working with binomial series Use properties of
Chapter 8, Problem 40E(choose chapter or problem)
Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series
\(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\).
\(\sqrt{4-16 x^{2}}\)
Questions & Answers
QUESTION:
Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series
\(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\).
\(\sqrt{4-16 x^{2}}\)
ANSWER:Solution 40EStep 1:In this problem we need to find the first four nonzero terms of the taylor series centered at ‘0’ for the function by using the taylor series : = 1+- +-............., for -1 < x , and interval of convergence of the new series.Consider , = = 2 ) , since = 2By , using for -1 < x we get = 2 )