Radius of convergence of $\sum \frac{n! x^{2n}}{n^n (1 + x^{2n})}$ What is the radius of convergence of
$$\sum_{n=1} \frac{n! x^{2n}}{n^n (1 + x^{2n})}$$
,which does not seem to be power series?
Ratio test gives me 
$$x^2 \lim_{n\to\infty}\frac{n^n(1+x^{2n})}{(n+1)^n (1+x^{2(n+1)})}$$
Root test gives me 
$$\lim_{n\to\infty}\frac{x^2\sqrt[n]{n!}}{n\sqrt[n]{1+x^{2n}}}$$
But both looks intimidating and I am having trouble evaluating the limits.

The ratio looks as follows:
$$x^2 \lim_{n\to\infty}\frac{n^n(1+x^{2n})}{(n+1)^n (1+x^{2(n+1)})}=x^2\lim_{n\to\infty}\frac{\frac{1+x^{2n}}{1+x^{2(n+1})}}{\frac{(n+1)^n}{n^n}}=x^2\lim_{n\to\infty}\frac{\frac{1+x^{2n}}{1+x^{2(n+1})}}{\frac{(n+1)^n}{n^n}}=x^2\frac{\frac{1}{x^2}}{e}=\frac{1}{e},{for}\ \mid x\mid>1$$
$$...=x^2\lim_{n\to\infty}\frac{\frac{1+x^{2n}}{1+x^{2(n+1})}}{\frac{(n+1)^n}{n^n}}=x^2\frac{1}{e}, for \ 0\le\mid x\mid\le1 $$
