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can you please help me with this problem? I have to analyze the convergence of the following series

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

thanks in advance.

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Welcome to Math.SE! Please, consider updating your question to include what you have tried and where you are getting stuck. That way, people on this site will know exactly what help you need. –  Did Oct 4 '13 at 16:08
    
There are some questions where, if you don't know the way, you can do almost nothing. This is, I believe, one of them. I'm voting for re-opening. –  DonAntonio Oct 4 '13 at 20:21

1 Answer 1

The seres converges.

In this question it is proved that $$ \sum_{n=1}^\infty\frac{\sin(n^2)}{n} $$ converges (warning: it is not easy.) Now $$ \sum_{n=1}^\infty\frac{\sin(n^2)}{\sqrt{n^2+1}}=\sum_{n=1}^\infty\frac{\sin(n^2)}{n}-\sum_{n=1}^\infty\sin(n^2)\Bigl(\frac{1}{n}-\frac{1}{\sqrt{n^2+1}}\Bigr). $$ The last series is absolutely convergent, since $$ \Bigl|\sin(n^2)\Bigl(\frac{1}{n}-\frac{1}{\sqrt{n^2+1}}\Bigr)\Bigr|\le\frac{\sqrt{n^2+1}-n}{n\sqrt{n^2+1}}=\frac{1}{n\sqrt{n^2+1}(\sqrt{n^2+1}+n)}\le\frac{1}{2\,n^3}. $$

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