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Solution: Series to functions Find the function represented
Chapter 8, Problem 54E(choose chapter or problem)
Series to functions Find the function represented by the following series and find the interval of convergence of the series.
\(\sum_{k=1}^{\infty} \frac{x^{2 k}}{4 k}\)
Questions & Answers
QUESTION:
Series to functions Find the function represented by the following series and find the interval of convergence of the series.
\(\sum_{k=1}^{\infty} \frac{x^{2 k}}{4 k}\)
ANSWER:Solution 54EStep 1:In this problem we have to find the function represented by the seriesand also we have to find the interval of convergence of the series.We compute the derivative of this power series by differentiating term-by-term: (Since ) where we’ve computed the sum by summing a geometric series with common ratio , so that the last series converges whenever and diverges when . That means that the interval of convergence of the last series is (1, 1) with or without the endpoints.