Convergence for $\sum_{k=1}^{\infty}(1-\cos(\frac{1}{\sqrt{k}}))$
Solution 1:
$$1 - \cos \left( {\frac{1}{{\sqrt k }}} \right) = 2\sin ^2 \left( {\frac{1}{{\sqrt k }}} \right)$$
Now using Limit comparison test with $\frac{1}{k}$ we get
$$\lim_{k\to\infty}\frac{2\sin^{2}\left(\frac{1}{\sqrt{k}}\right)}{\frac{1}{k}}=2$$.
$\sum_{k=1}^{\infty}\frac{1}{k}$ diverges . Hence By limit comparison test . The original series also diverges.