I have a task to research $$\sum\limits _{n=1}^{\infty}\left(\frac{x\sin\frac{x}{\sqrt{n}}}{x^{3}+n}\right)^{2}$$ for uniform convergence on $0<x<+\infty$. I tried using Weierstrass M-test, but couldn't find the sequence. So what test should I use to prove uniform or non-uniform convergence of this series?
2026-04-07 15:00:03.1775574003
Uniform convergence of $\sum\limits _{n=1}^{\infty}\left(\frac{x\sin\frac{x}{\sqrt{n}}}{x^{3}+n}\right)^{2}$
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Hint:
$$\sup_{x \in (0, \infty)}\frac{x^2}{(x^3 +n)^2} = O\left(n^{-4/3} \right)$$