Pointwise and uniform convergence of series $\sum_{n=1}^{\infty} \frac{\sin nx}{n^2}$ and $\sum_{n=1}^{\infty} \frac{(-1)^n}{x^2+n}$

Question

For which domains in R the series of functions $\displaystyle \sum_{n=1}^{\infty} \dfrac{\sin nx}{n^2}$ and $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n}{x^2+n}$ converge pointwise and for which domains do they converge uniformly. I'm not sure if they would be different for uniform and pointwise.

Answer 1

For the second series $S(x)=\sum_{n=1}^\infty \frac{(-1)^n}{n+x^2}$, we note that for all $N$, the partial sum $\left|\sum_{n=1}^N (-1)^n\right|$ is bounded by $1$ for all $N$.

Furthermore, we see that $\frac{1}{n+x^2}$ is monotonic and uniformly converges to $0$ since for all $\epsilon >0$, we have

$$\left|\frac{1}{n+x^2}\right|\le \frac1n <\epsilon$$

whenever $n>1/\epsilon$ for all $x$.

Therefore, Dirichlet's Test for Uniform Convergence guarantees that $S(x)$ converges uniformly. And uniform convergence trivially implies pointwise convergence.

Answer 2

The only case I can find for the first serie converging is if $x = mod(\pi)$.

The second serie $\sum {(-1)^n\over x^2+n}$ is an alternate serie (ie : of the form $\sum (-1)^n a_n$ ) thus is converges if the sequence $1\over x^2+n$ is positive decreasing and its limit is $0$ which is the case $\forall x \in \mathbb{R}$