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?
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$\begingroup$ I assume what you mean is you are investigating the uniform convergence of the sequence (as $k\to\infty$) of functions $$f_k(x) = \sum_{n=1}^{k}\left( \frac{x\sin\frac{x}{\sqrt{n}}}{x^3+n} \right)^2$$. $\endgroup$– Mark FischlerDec 19, 2016 at 5:49
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1 Answer
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Hint:
$$\sup_{x \in (0, \infty)}\frac{x^2}{(x^3 +n)^2} = O\left(n^{-4/3} \right)$$
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$\begingroup$ @Henrik: Since when are hints prohibited in lieu of complete answers? If so then this has never been applied consistently for even a tiny percentage of the time. $\endgroup$– RRLDec 19, 2016 at 7:50
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$\begingroup$ Sorry, I was too quick. $\endgroup$ Dec 19, 2016 at 7:55