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

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.
