During revision, I came across this problem:
The set of real numbers $x$ for which the series $\sum_{n=1}^{\infty}{\frac{n!x^{2n}}{n^n(1+x^{2n})}}$ converges is __.
I tried using the ratio test, but got stuck in the process of simplification.
(The answer is the series converges for $\mathbb{R}$.)
Sincere thanks for any help.