Determine the values of x ∈ R for which the following series converge:
$\sum_1^\infty \frac{x^nn^n}{n!}$
My attempt: I used the ratio test to obtain $|x|< \frac{1}{e}$ as the interval where it convergences for sure, but I am unable to determine whether the series convergences at $x=\frac{1}{e}$ and $x=\frac{-1}{e}$ (when the limit of the ratio is $1$ and the test is inconclusive).
Best Answer
For the cases $x=\pm \frac 1 e$ we can use Stirling approximation
$$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$