Find the domain of convergence of the series $\sum^{\infty}_{n=1}\frac{n!x^{2n}}{n^n(1+x^{2n})}$

Question

Find the domain of convergence of the series $\sum^{\infty}_{n=1}\frac{n!x^{2n}}{n^n(1+x^{2n})}$.

Using the ratio test, I got $ \left | \frac{x+x^{2n+1}}{1+x^{2n+1}} \right | $, but I don't know how to proceed from there.

Any help would be appreciated!

Answer 1

The summand $\displaystyle\frac{n!}{n^n}\frac{x^{2n}}{1+x^{2n}}$ is always nonnegative and less than or equal to $\displaystyle\frac{n!}{n^n}$. Furthermore, the series $\displaystyle\sum_{n=1}^\infty \frac{n!}{n^n}$ converges by the ratio test. Therefore the series $\displaystyle\sum_{n=1}^\infty \frac{n!}{n^n}\frac{x^{2n}}{1+x^{2n}}$ converges for all real numbers $x$.