I'm trying to use the limit comparison test to test for convergence but I've got different results depending on what I use as my $b_k$
If I use $b_k=\frac{1}{k^{2}}$, I have $\frac{a_k}{b_k}\to 1$ $\sum_{k=1}^{\infty}b_k$ converges, so $\sum_{k=1}^{\infty}a_k$ will also converge
But if I use $b_k=\sin\frac{1}{k}$, $\frac{a_k}{b_k}\to 0$. And here, $b_k$ diverges, $a_k$ must diverge also...
Help?