Does this series converge pointwise or uniformly on $\Bbb R$?

Question

I am given the following fourier-series:

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\,\sin(nt)$$

I want to figure out if the series converges uniformly or pointwise on $\Bbb R$, and if converges pointwise for $t=\frac{\pi}{2}$:

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\,\sin\left(\frac{\pi}{2}n\right)$$

Can I use the Weierstrass-M-Test here?

$$\left\lvert\frac{(-1)^n}{n}\sin\left(\frac{\pi}{2}n\right)\right\rvert \le \left\lvert\frac{1}{n}\right\rvert$$

and since the the series $\sum_{n=0}^{\infty}\frac{1}{n}$ diverges my series is not uniformly convergent.

Answer 1

Your series is the Fourier series of the function $f(x)=-\frac{x}{2}$ over $(-\pi,\pi)$, extended by periodicity. The proof is straighforward, you just have to compute $$ \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)\,dx $$ through integration by parts. Then the situation is the following:

$\hspace1in$

with pointwise convergence for every point of $I=(-\pi,\pi)$.

In virtue of Gibbs' phenomenon, the convergence on $I$ is not uniform.