Find the values $x \in\mathbb{R}$ where series $\sum^\infty_{n=1}\frac{x^n}{1+x^n}$ converges. [duplicate]

I'm trying to find the values $x \in\mathbb{R}$ where series $\sum^\infty_{n=1}\frac{x^n}{1+x^n}$ converges.

I'm pretty sure that only values where $x\in(0,1)$ allow the series to converge because values within that range will shrink to 0 as n approaches infinity. Any value larger than 1 will create increasingly large summands and the series will eventually reach infinity.

Solution 1:

I am answering for $x\geq 0$. For $x\in [0,1)$, we have $$\frac{x^n}{1+x^n} \leq x^n,$$ and the series $\sum_{n=1}^\infty x^n$ converges, so indeed the initial series converge. For $x> 1$, $x^n/(1+x^n)\to 1>0$ and so the series diverges. For $x=1$, $x^n/(1+x^n)=1/2$ and the series diverges as well.