## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 11 - Infinite Series - 11.6 Power Series - Exercises - Page 577: 17

#### Answer

The series converges for all $x$ and the radius of converges is $R=\infty$

#### Work Step by Step

Apply the ratio test: \begin{aligned} \rho &=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|\frac{(2(n+1)) ! x^{n+1}}{((n+1) !)^{3}} \cdot \frac{(n !)^{3}}{(2 n) ! x^{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|x \frac{(2 n+2)(2 n+1)}{(n+1)^{3}}\right| \\ &=\lim _{n \rightarrow \infty}\left|x \frac{4 n^{2}+6 n+2}{n^{3}+3 n^{2}+3 n+1}\right|\\ &=\lim _{n \rightarrow \infty}\left|x \frac{4 n^{-1}+6 n^{-1}+2 n^{-3}}{1+3 n^{-1}+3 n^{-2}+n^{-3}}\right|=0 \end{aligned} Then the series converges for all $x$ and the radius of converges is $R=\infty$

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