$\sum^{\infty}_{n=1} \frac {x^n}{1+x^{2n}}$ interval of convergence I need to find interval of convergence for this series:
$$\sum^{\infty}_{n=1} \frac {x^n}{1+x^{2n}}$$
I noticed that if $x=0$ then series is convergent. Unfortunately, that’s it.

If $\left| x\right|<1$ then
$$
\left| \frac{x^n}{1+x^{2n}} \right|\leq \left| x\right|^n
$$
because $1+x^{2n}\geq1$. Now $\sum_{n=1}^{\infty} \left| x\right|^n$ is a finite geometric series.
Similarly, if $\left| x\right|>1$ then
$$
\left| \frac{x^n}{1+x^{2n}} \right|\leq \frac{1}{ \left| x\right|^n}
$$
because $1+x^{2n}\geq x^{2n}$. Now $\sum_{n=1}^{\infty} \frac{1}{\left| x\right|^n}$ is a finite geometric series.
If $x=1$, or $x=-1$ the series is divergent.
