Uniform convergence $\sum\limits _{n=1}^{\infty}\frac{\cos\frac{x}{n}\arctan\frac{x}{\sqrt{n}}}{nx+x^{2}}$

Question

I' ve got a task to research $\sum\limits _{n=1}^{\infty}\frac{\cos\frac{x}{n}\arctan\frac{x}{\sqrt{n}}}{nx+x^{2}}$ for a uniform convergence on $0<x<+\infty$. I tried to use the Weierstrass M-test, but I wasn't able to find the number sequence. So the question is: how to I show if this series converges uniformly or not?

Answer 1

Hint:

Observe that for $\;x>0\;$ we have that $\;\arctan x\le x\;$ , so

$$\left|\frac{\cos\frac xn\arctan\frac x{\sqrt n}}{nx+x^2}\right|\le\frac{1\cdot\frac x{\sqrt n}}{xn}=\frac1{n^{3/2}}$$