Suppose that the series $\sum_{1}^{\infty}n|b_n|$ converges. Show that the series $\sum_{1}^{\infty}b_n \sin{nx}$ converges uniformly on $\mathbb{R}$, and that it can be integrated and differentiated term by term.
So I have to use the Weierstrass M-test. Put $M_n=\sup_{x \in \mathbb{R}}|b_n \sin{nx}|=|b_n|$. Now we want to show that $\sum_{1}^{\infty}M_n =\sum_{1}^{\infty}|b_n|$converges then $\sum_{1}^{\infty}b_n \sin{nx}$ converges uniformly.
Do I need to bound $\sum_{1}^{\infty}M_n$ by $\sum_{1}^{\infty}n|b_n|$? I understand the that it can be integrated and differentiated term by term if $\sum_{1}^{\infty}b_n \sin{nx}$ converges uniformly on $\mathbb{R}$.