Uniform convergence of the series $\sum_{n=1}^{\infty}{\frac{x \sin(nx)}{n^2 x |\ln(x)| + \sqrt{n}}}$

Question

I am studying the series

$$\sum_{n=1}^{\infty}{\frac{x \sin(nx)}{n^2 x |\ln(x)| + \sqrt{n}}}$$

and I want to determine whether it converges uniformly on the interval $(0, \infty)$.

I have applied the Weierstrass M-test and established that the series converges uniformly on the intervals $(0, 1/e]$ and $[e, \infty)$ where ($|\ln(x)| \geq 1$) by estimating

$|f_n(x)| \leq \frac{1}{n^2}$.

However, I am unsure how to analyze the uniform convergence on the interval $(1/e, e)$.

Attempt to Analyze $(1/e, e)$

To further investigate the uniform convergence on the interval $(1/e, e)$, I considered breaking this interval into two subintervals: $(1/e, 1]$ and $(1, e)$. I attempted to apply the Dirichlet test for uniform convergence on these subintervals, but I faced challenges in proving the necessary monotonicity.

How can I approach the analysis of uniform convergence on $(1/e, e)$?

Thank you!

Answer 1

Let $a_n(x)=\frac{x}{n^2x|\ln x|+\sqrt n}$ and $b_{n}(x)=\sin(nx)$ for $x\in(1/e,e)$. We have the following.

1. The sequence $\{a_n(x)\}$ is decreasing for fixed $x\in(1/e,e)$ and uniformly convergent to $0$; in fact, we have $$0\leq a_{n}(x)\leq\frac{e}{\sqrt n}\to0.$$
2. The partial sum $\sum_{k=1}^{n}b_k(x)$ is uniformly bounded. See here for more details.

So, by Dirichlet's test, $\sum a_{n}(x)b_n(x)$ is uniformly convergent on $(1/e,e)$.