I'm working on a problem where I need to show that the series of functions, $$ f(x) = \sum_{n\geq 1} \frac{x^n}{n^2}, $$ converges to some $f(x)$, and that $f(x)$ is continuous, differentiable, and integrable on $[-1,1]$.
I know how to show that $f(x)$ is continuous, since each $f_n(x)$ is continuous, and $f_n(x)$ converges uniformly. Because each $f_n(x)$ is also integrable, I can also show $f(x)$ is integrable.
The trouble I'm having is proving that $f(x)$ is differentiable. I need to show that the series of derivatives converges uniformly. However, I don't think I can use the Weierstrass M-Test in this scenario. Any ideas?