Prove uniform convergence of $\sum_{n=1}^\infty (-1)^{m+1} \frac{2}{m\pi} \sin(m \pi x)$

Question

I am trying to use the Weierstrass M-test to show that the series $\sum_{m=1}^\infty (-1)^{m+1} \frac{2}{m\pi} \sin(m \pi x)$ converges uniformly on $[0,a]$, with $a<1$. (Here, we have that $0\leq x \leq 1$.

I know I need to find a convergent sequence $(M_m)$, where $\lvert (-1)^{m+1} \frac{2}{m\pi} \sin(m \pi x) \rvert = \lvert \frac{2}{m\pi} \sin(m \pi x)\rvert \leq M_n $. I tried $M_m = \frac{2}{m \pi}$. But that doesn't work since the sequence $(M_m)$ fails converge.

What is a better choice for the sequence $(M_m)$? It it matters, I got that series from a PDE book and the author claims it is uniformly convergent, and I am attempting to verify it.

Answer 1

Hint: Dirichlet test

Note that $(-1)^m \sin m\pi x= \cos m\pi \sin m\pi x = \sin( m \pi (x+1)).$