Show that $\sum_{n=1}^{\infty}\frac{\sin\frac{x}{n}\sin2nx}{x^2+4n}$ converges uniformly. 
How to show that the following series converges uniformly?
  $$
\sum_{n=1}^{\infty}u_n(x),\ \ \ u_n(x)=\frac{\sin\frac{x}{n}\sin2nx}{x^2+4n},\ \ 
x\in E=(-\infty;+\infty)
$$

At first I tried to apply Dirichlet's test. However, I got stuck while trying to prove that $\sum_{n=1}^{\infty}\sin\frac{x}{n}\sin2nx$ is less than some fixed $M$ (multiplying and dividing by $2\sin x$ did not help much). In my other attmepts I also got stuck trying to limit the numerator. So, the problem is with these $\sin$ functions.
 A: As $f(x)=\sum_{n=1}^{\infty}\frac{\sin\frac{x}{n}\sin2nx}{x^2+4n}$ is even, we can limit the analysis on $[0, \infty)$.
Consider $$v_n(x)=\frac{x}{n(x^2+4n)}.$$
We have
$$v_n^\prime(x)=\frac{4n^2-nx^2}{n^2(x^2+4n)^2}$$
Based on that, one can prove that $v_n$ is positive on $[0,\infty)$ and attains its maximum at $x_n = 2\sqrt n$. The maximum having for value $\frac{1}{2n^{3/2}}$. As $\sum \frac{1}{n^{3/2}}$ converges, $\sum v_n(x)$ converges uniformly on $[0, \infty)$ according to Weierstrass M-test.
We then get the uniform convergence of $\sum u_n(x)$ as $\vert u_n(x) \vert \le v_n(x)$ for all $n \in \mathbb N$ for $x \in [0,\infty)$.
A: We have the following inequality:
$$\left|\frac{\sin\frac{x}{n}\sin 2nx}{x^2+4n}\right| \leq \frac{|x|}{nx^2+4n^2}$$
Then from calculus optimization, we have that the max of the term on the right occurs at $x=\pm2\sqrt{n}$ so we get an even further inequality
$$\frac{|x|}{nx^2+4n^2}\leq \frac{1}{4n^{\frac{3}{2}}}$$
The series is less than $\sum \frac{1}{4n^{\frac{3}{2}}}$, which is independent of $x$, so the convergence is uniform.
A: First apply Abel's uniform convergence test: series
$$
\sum_n a_n(x) b_n(x)
$$
converges uniformly if
$$
\sum_n a_n(x) \text{ converges unifromly}
$$
and
$$
b_n(x) \text{ is monotone and uniformly bounded sequence}.
$$
Indeed, $b_n(x) = \sin \frac{x}{n}$ is monotone sequence for each $x$ and it's uniformly bounded. The only thing left is to establish uniform convergence of this series:
$$
\sum_{n} \frac{\sin 2nx}{x^2 + 4n}
$$
It can be done with Dirichlet's test.
