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Representing functions by power series
Chapter 8, Problem 54E(choose chapter or problem)
Representing functions by power series Identify the functions represented by the following power series.
\(\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{k+1}}{4^{k}}\)
Questions & Answers
QUESTION:
Representing functions by power series Identify the functions represented by the following power series.
\(\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{k+1}}{4^{k}}\)
ANSWER:Solution 54EStep 1:In this problem we need to identify the function represented by the power series Consider , = ++++........... = -+-+........... Is an infinite geometric series , here “ the first term (a) = x , and the common ratio (r)= = We know that a+ar+ Then () = , if |r|< 1 , , if |r| > 1 Therefore , -+-+........... . is an infinite geometric series, then = = = = f(x) . Therefore , the function f(x) = represented by the power series