# Chapter 11 - Review - Exercises - Page 786: 52

The series converges for all values of $x$. The interval of convergence is $R$ and radius of convergence is $\infty$.

#### Work Step by Step

$10^{x}=\Sigma_{n=0}^\infty(x)^{n}\frac{(ln10)^{n}}{n!}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{(ln10)^{n+1}x^{n+1}}{n+1!}}{\frac{(ln10)^{n}x^{n}}{n!}}|$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{xln10}{n+1}|$ $=0$ $=0 \lt 1$ Thus, the series converges for all values of $x$. The interval of convergence is $R$ and the radius of convergence is $\infty$.

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