# Determine the radius of convergence of the series $\sum \limits_{n=0}^\infty (10^{−n} + 10^n)x^n$

Determine the radius of convergence, R, of the power series $\sum \limits_{n=0}^\infty a_nx^n$ where $a_n = 10^{−n} + 10^n$.
I know the series diverges and the radius of convergence is 0 but how do I prove it?

• How do you know that the series diverges? How do you know that the radius of convergence is $0$? Have you tried $x=\frac{1}{100}$? Mar 28 '16 at 19:26

$$\lim_{x\rightarrow\infty}\left|\frac{a_{n+1}x^{n+1}}{a_nx^n}\right|=\lim_{n\rightarrow\infty}\left|\frac{(10^{-n-1}+10^{n+1})x^{n+1}}{(10^n+10^{-n})x^n}\right|=\lim_{n\rightarrow\infty}\left|\frac{(10^{-n-1}+10^{n+1})x}{(10^n+10^{-n})}\right|$$
For what values of $x$ is this limit less than $1$? To make this easier, we can divide the numerator and denominator by $10^n$ to get that this limit is the same as:
$$\lim_{n\rightarrow\infty}\frac{(10^{-2n-1}+10)|x|}{(1+10^{-2n})}.$$
As $n$ increases, the limit approaches $10|x|$, so we can find appropriate $x$'s.
The series does not have the radius of convergence $0,$ but rather $1/10.$