why is $\sum_{n=1}^{\infty} \frac{1}{n} \sin\left( \frac{x^n}{\sqrt n} \right)$ uniformly convergent? [duplicate]
Solution 1:
Use Weirestrass M test. We have $|\sin(x)|\leq |x|,\,\forall x\in\mathbb{R}$
$\displaystyle|\frac{1}{n} \sin\left(\frac{x^n}{\sqrt n}\right)|\leq|\frac{1}{n}||\frac{x^{n}}{\sqrt{n}}|\leq\frac{1}{n}\frac{1}{\sqrt{n}}$. This is because $x\in[-1,1]$
Since $\sum_{n=1}^{\infty}\frac{1}{n\sqrt{n}}$ converges you have uniform convergence by Weirestrass-M Test .