Examine the continuity and differentiability of functions:
a) $\displaystyle f(x)=\sum_{n=1}^{+\infty}\frac{\sin(nx)}{n^3}$
b) $\displaystyle f(x)=\sum_{n=1}^{+\infty}\arctan\left(\frac{x}{n^2} \right)$
in the case of differentiability explore the sign of f '(0).
So, we are dealing with function series I think. I tried a): Let $f_n(x)=\frac{\sin(nx)}{n^3}$. With Weierstrass M-test $|f_n|\le \frac{1}{n^3}$ so the series $\sum_{n=1}^{+\infty}f_n$ converges uniformly. The same applies for series of derivatives that is: $\sum_{n=1}^{+\infty}\frac{\cos(nx)}{n^2}$ so function $f$ is differentiable and so it is continuous. f'(0)>0
Is this argumentation ok?
I don't know how to approach b).