interval of convergence of $\sum_{n=0}^{\infty} (nx)^{n}/n! $

Question

Im not sure how to go about finding the interval of convergence of $\sum_{n=0}^{\infty} (nx)^{n}/n! $

I think i remember learning that you can either use the ratio test or cauchy root test to solve it. I have tried both and gotten nowhere significant.

Answer 1

Notice that radius of convergence $=lim|\frac{a_n}{a_{n+1}}|$

Hence for your series, radius of convergence $=lim\frac{n}{n+1}^n$ after simplify the expression.

And we get the limit by the following steps,

$\frac{n}{n+1}^n=\frac{1}{(1+\frac{1}{n})^n}$

taking $n\to\infty$ the above expression becomes $\frac{1}{e}$.

Hope this would help you.