The 10 standard tests taught in class are:
1) $n^{th}$ term test for divergence.(Not applicable: $\lim =0$).
2) Geometric Series(Not applicable).
3) Telescoping Series(Not applicable)
4) Integral Test(Not applicable: $f<0$ sometimes)
5) $p$-series(Not applicable)
6) Direct Comparison(maybe)
7) Limit Comparison(Not applicable $a_n<0$ sometimes)
8) Alternating Series Test(Not Alternating)
9) Ratio Test fails
10) Root Test fails
I did find a hint online that states we should show that for $k^2+1\leq n\leq k^2+k$ we have $\sum\limits_{n=k^2+1}^{k^2+k}\frac{\sin(\sqrt{n})}{\sqrt{n}}>\frac{1}{8}$. Is there an easier way and if not how should we go about showing this?