Convergence of the following sum Does the following sum converge? $$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{n}$$
I tried the ratio test and got that $\rho=0$ which means that the series converges absolutely. However, Mathematica and Wolfram Alpha do not give a result when trying to find its convergence. Am I wrong? 

WA is obviously incorrect.  To show this, recall that $\sin^2(n)=\frac12-\frac12\cos(2n)$.  
Then, letting $S_N=\sum_{n=1}^N \frac{\sin^2(n)}{n}$, $H_N=\sum_{n=1}^N\frac1n$, and $T_N=\sum_{n=1}^N\frac{\cos(2n)}{n}$, it is easy to see that
$$\begin{align}
S_N+\frac12 T_N
&=\frac12 H_N\end{align}$$
Note that Dirichlet's Test guarantees that $T_N$ converges.  Hence, if we assume that $S_N$ converges, then the sum $S_N+\frac12 T_N=H_N$ converges.
Inasmuch as the Harmonic series diverges, this leads to a contradiction and we find that $S_N$ diverges.  And we are done. 
