How to prove that $$1+ \sum_{n=1}^{+\infty}{x^n \over n(n+1)}$$ converges uniformly in $[-1,1]$?
I managed to prove it in $[0,1]$, I know that it converges pointwise in $[-1,1]$ and it's a polynomial so it's always continuous and the derivative is continuous as well, and I also know it's $\le 1$. Help?
We have $$\sum_{n=1}^{\infty}{|x^n| \over n(n+1)}\le \sum_{n=1}^{\infty}{1 \over n(n+1)} <\infty,$$ so the Weiestrass M-test applies.