<s>[INST] $\sum_{k=2}^\infty \frac{1}{1+(-1)^k \sqrt k}$ converges or diverges? $\frac{1}{1+\sqrt k (-1)^k}= \frac{1-(-1)^k \sqrt k}{1-k}$. Not quite sure how to move on from here.
 [/INST] Rewrite
\begin{align}\sum_{k=2}^{2N+1}\frac{1}{1+(-1)^k \sqrt k}&= \sum_{k=1}^N\left(\frac{1}{1+(-1)^{2k} \sqrt {2k}}+\frac{1}{1+(-1)^{2k+1} \sqrt {2k+1}}\right)\\&=\sum_{k=1}^N\left(\frac{1}{1+\sqrt {2k}}+\frac{1}{1- \sqrt {2k+1}}\right)\\&=\sum_{k=1}^N\left(\frac{1- \sqrt {2k+1}+1+\sqrt {2k}}{(1+\sqrt {2k})(1- \sqrt {2k+1})}\right)
\\&<\sum_{k=1}^N\left(\frac{2}{(1+\sqrt {2k+1})(1- \sqrt {2k+1})}\right)
\\&=\sum_{k=1}^N\left(\frac{2}{-2k}\right)\to-\infty
\end{align}
</s>