# Analyze the convergence of the series

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

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

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 $$\Bigl|\sin(n^2)\Bigl(\frac{1}{n}-\frac{1}{\sqrt{n^2+1}}\Bigr)\Bigr|\le\frac{\sqrt{n^2+1}-n}{n\sqrt{n^2+1}}=\frac{1}{n\sqrt{n^2+1}(\sqrt{n^2+1}+n)}\le\frac{1}{2\,n^3}.$$