Uniform convergence of $ \sum_{n=1}^\infty \frac{x\cdot\sin\sqrt {x/n}} {n + x} $

Question

I need to check on the uniform convergence. $$ \sum_{n=1}^\infty \frac{x\cdot\sin\sqrt {\frac x n}} {n + x} $$

on the interval:

a) $\; (0, 1) $

b) $\; (1, +\infty) $

I think I need to use Cauchy ratio, but I cannot understand how

Answer 1

For part $(a)$:

$$\begin{align} \left|\sum _{n=1}^\infty \frac{x\sin {\sqrt {\frac{x}{n}}}}{n+x}\right| &\le \sum _{n=1}^\infty \left|\frac{x\sin {\sqrt {\frac{x}{n}}}}{n+x}\right|\\\\ &\le \sum_{n=1}^\infty \frac{\left|x\,\sqrt{\frac{x}{n}}\right|}{n+x}\,\,\text{since} |\sin x|\le |x|\\\\ &\le \sum_{n=1}^\infty \left|\frac{x^{\frac{3}{2}}}{n^{\frac{1}{2}}(n+x)}\right|\\\\ &\le \left|\sum_{n=1}^\infty \frac{x^{\frac{3}{2}}}{n^{\frac{3}{2}}}\right|\\\\ &\le \sum_{n=1}^\infty \frac{1}{n^{\frac{3}{2}}} \to 0 \end{align}$$

provided $|x|<1$

By Weistrass M Test we have the series converges uniformly.