I want to show whether the following series converges pointwise? and whether it converges uniformly? $\sum_{n=1}^\infty \frac{arctan(nx)}{n^3} $ on $\mathbb{R}$.
For pointwise, I'm not sure how to approach it, can I have some hint.
For uniform convergence I used the M-test: $ |\frac{arctan(nx)}{n^3}| \leq \frac{0.5\pi}{n^3}$ and $\sum_{n=1}^\infty \frac{0.5\pi}{n^3} < \infty $. So it's uniformly convergent. Is this correct? If it does uniformly converge then it converges pointwise. Is there a way to check whether it converges pointwise without knowing it's uniformly convergent?