Uniform convergence of $\sum\limits_{n=1}^\infty\frac{\sin\frac{x}{n}\sin 2nx}{x^2+4n}$

Question

I have a task to research $\sum\limits_{n=1}^\infty\frac{\sin\frac{x}{n}\sin 2nx}{x^2+4n}$ for uniform convergence on $ -\infty<x<+\infty $. I think, I should use Weierstrass M-test, but I can't find the sequence. So what sequence should I use or maybe another test?

Answer 1

Let $f_n(x)$ be given by

$$f_n(x)=\frac{\sin(x/n)\sin(2nx)}{x^2+4n}$$

Using the AM-GM inequality, we assert that

$$|f_n(x)|\le \frac{|x|}{n(x^2+4n)}\le \frac{1}{4n^{3/2}}$$

Inasmuch as $\displaystyle \sum_{n=1}^{\infty}\frac{1}{4n^{3/2}}<\infty$, the series $\displaystyle \sum_{n=1}^\infty f_n(x)$ uniformly converges for $x\in \mathbb{R}$ and not only on compact subsets thereof.

Answer 2

Hint. One may observe that $$ \left|\frac{\sin\frac{x}{n}\sin 2nx}{x^2+4n}\right|\le\frac{\left|\sin\frac{x}{n}\right|}{4n}\le \frac{\frac{\left|x\right|}n}{4n}\le\frac{\left|A\right|}{4n^2},\quad x \in [-A,A] \,\,(A>0), $$ by using the Weierstrass M-test, this gives the uniform convergence of the given series over each compact set of $\mathbb{R}$.