Show that $\sum_{n \in \mathbb{N}} \frac{(-x)^n}{n(1 + x^n)}$ converges uniformly for all $x \in \mathbb{R}^+$

I have tried to use abel's test and solve as following

Take sequence ${b_n(x)}$ as $\frac{-1^n}{n}$ which is uniformally convergent by Lebinitz test. Also $\frac{x^n}{x^n+1}$ = $\frac{1+x^n-1}{1+x^n}$ = $1 - \frac{1}{1+x^n}$ but this is not monotonic decreasing for a particular value of $x$. How to proceed further?