Weierstrass uniform convergence - Stuck to the point.

Question

$$\sum_{n=1}^\infty \frac{\sin (nx)}{n!}$$ Interval: $x \in(- \infty, + \infty)$

I've been trying to do this all day, but I just cant get to the end of it. It's not that I do not understand the point, but I'm having issues doing it this particular way. I'd be grateful if somebody would show me how to find uniform convergence with criteria.

Thank you!

Answer 1

For all $\;x\in\Bbb R\;$ ,we have

$$\left|\frac{\sin nx|}{n!}\right|\le\frac1{n!}\implies\;\text{since}\;\;\sum_{n=1}^\infty\frac1{n!}$$

converges (to $\;e-1\;$ , by the way), Weierstrass M test gives absolute convergence in the whole real line.