Radius of convergence of $1+3x+\frac{3^2x^2}{2!}+\cdots$

Question

The question is to find the radius of convergence of the power series $1+3x+\frac{3^2x^2}{2!}+\frac{3^3x^3}{3!}+\cdots$

The answer is given to be $\frac{1}{3}$

My attempt:

$a_n=\frac{3^n}{n!}$

So, using Ratio Test,

$|\frac{a_{n+1}}{a_n}|=|\frac{3}{n+1}|$

So, $lim|\frac{a_{n+1}}{a_n}|=lim\frac{3}{n+1}=0$

Hence the power series is everywhere convergent. Where am I going wrong? Please help!

Answer 1

You're right. This series defines function:

$$e^{3x}=\sum_{n=0}^{\infty}\frac{3^nx^n}{n!}$$

So it's convergent everywhere.There's something wrong with answer $\frac{1}{3}$.