Hi there. I have a problem with uniform convergence of functional series.

Question

I have no idea what to do and how to start even, please give some pieces of advice. And if you can solve this task I will be very thankful $$\sum_{n=1}^{\infty}(\frac{\sin(nx)}{\sqrt[]{n^{3}+1}},\quad x\in (-\infty; \infty) $$

Answer 1

Hint. Note that for any $x\in \mathbb{R}$, $$\frac{|\sin(nx)|}{\sqrt{n^{3}+1}}\leq \frac{1}{n^{3/2}}.$$ Then use Weierstrass M-test.

Answer 2

We know that

$$(\forall n\in \mathbb N)\;(\forall x\in\mathbb R)\;\;\;|\sin (nx)|\leq 1$$

$$\implies (\forall n>0) \; (\forall x\in \mathbb R)$$ $$\;\;|u_n (x)|\leq \frac {1}{n^{3/2}} $$

$$\implies (\forall n>0) \;\;\sup_{\mathbb R}|u_n(x)|\leq \frac {1}{n^{3/2}} $$ the series $\sum u_n (x) $ is normally and then uniformly convergent at $\mathbb R$.