${S_k} $ is the partial sum sequence of $ \sum_{n=0}^\infty a_n$,
Given $\lim_{k \rightarrow \infty}S_{2k}$ exists and $\lim_{n \rightarrow \infty}a_{n}=0$, can you prove that $ \sum_{n=0}^\infty a_n$ converges?
My hunch is yes, but I get kind of stuck
to prove $ \sum_{n=0}^\infty a_n$ converges, we need to show that $$\lim_{k \rightarrow \infty}S_{2k+1} = \lim_{k \rightarrow \infty}S_{2k}$$
we know $\lim_{n \rightarrow \infty}a_n = 0$, so $\lim_{n \rightarrow \infty}2a_n = 0$ and $\lim_{n \rightarrow \infty}a_{2n+1} = 0$
I dont really know where to go from here, so any pointers would be appreciated, thank you!