Radius of Convergence of Infinite Series I am trying to find the radius of convergence of the series
$$
1+2x+\frac{3^2x^2}{2!}+\frac{4^3x^3}{3!}+\ldots
$$
I found the general term $(\sum_{n=0}^{\infty}\frac{(n+1)^{n}}{n!}x^{n})$, and I know that the radius of convergence is $\frac{1}{e},$ but I haven't been able to show this. I've tried to use the ratio test, but I haven't found the solution.

Use the ratio test:
$$ \frac{\frac{(n-1+1)^{n-1}}{(n-1)!}}{\frac{(n+1)^{n}}{n!}}=\frac{n^n}{(n+1)^n}=\frac1{\left(1+\frac1n\right)^n}$$
and
$$ \left(1+\frac1n\right)^n\to e$$
