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I can't prove or refute uniform convergence of $ \sum_{n=1}^{n=+\infty} \frac{\sqrt{x}}{x^2+n}\sin(\frac{n}{x}),x>0$

I tried to use Abel's test, but I haven't got any idea how to prove сonvergence of $\sin(n/x) $

I tried : $\sin(1/x)+..+sin(n/x)=Im(e^{i\frac{1}{x}}+..+e^{i\frac{n}{x}})=Im(e^{i-\frac{1}{x}}\frac{e^{i\frac{n}{x}}-1}{e^{i\frac{1}{x}}-1})\leq \frac{1}{|e^{i\frac{1}{x}}-1|}$

Mike Terentyev
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  • Can you show us what you have attempted? Or what is interesting about this question? Usually questions posed with the tone "do this homework problem for me" are not well-received. – Mason Nov 26 '18 at 20:09
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    @Mason I tried to use Abel's test, but I haven't got any idea how to prove сonvergence of sin(n/x) – Mike Terentyev Nov 26 '18 at 20:20
  • Related: https://math.stackexchange.com/questions/13490/proving-that-the-sequence-f-nx-sum-limits-k-1n-frac-sinkxk-is – Mason Nov 26 '18 at 22:08

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