Given the series ${a_n}$ : $a_n= \sum _{k=1}^{n}{\frac { \left( -1 \right) ^{k}}{{3}^{k}+\sqrt {k}}}$ prove that ${a_n}$ converges.
So far this is what I've done: I've split this into 2 sub-series: $k=2n$ and $k=2n+1$
for $(k=2n)$ I get: $\sum _{k=1}^{n} \left( {3}^{k}+\sqrt {k} \right) ^{-1} = 0$ (as k goes to infinity the whole series converges to 0)
for $(k=2n+1)$ I get:
$\sum _{k=1}^{n}- \left( \left( {3}^{k} \right) ^{-1}+\sqrt {-k} \right) ^{-1}$ (which also converges to 0 as k approaches to infinity)
Since both subs-series of $a_n$ converge to the same value, $0$, $a_n$ converges, in particular to $0$.
I'm not sure if this is a tight enough proof since maybe by chance it works out here specifically. I also checked separate limits as k approaches $-\infty$. They both converge to $0$ for each sub-series. What am I missing? Thanks.