Question

For which real \(x\) is the series \[\sum_{n=1}^{\infty}\frac{x^{n}}{1+x^{2n}}\]convergent? For which intervals \(I\) is the series uniformly convergent on \(I\)?

Answer 1

Since \(\frac{x^n}{1+x^{2n}}<\frac{1}{x}\), we know that \(\frac{x^n}{1+x^{2n}}\) converges

Answer 2

\[
\sum \frac 1 x
\]Isn't a convergent series.