More generally:
Let $(x_n)_{n\geqslant1}$ denote a bounded sequence, $s_n=\sum\limits_{k=1}^nx_k$ and $u_n=\tfrac1ns_n$. Then $(u_n)_{n\geqslant1}$ converges (to a limit $\ell$) if and only if $(u_{n^2})_{n\geqslant1}$ converges (to the same limit $\ell$).
Only one direction needs proof, hence one assumes that $(u_{n^2})_{n\geqslant1}$ converges. One can assume without loss of generality that $|u_n|\leqslant C$ for every $n$ and that $\ell=0$. Then, for every $n\geqslant1$, there exists $k\geqslant1$ such that $k^2\leqslant n\lt (k+1)^2$, and $ |u_n|\leqslant\tfrac1n|s_{k^2}|+2kC\tfrac1n\leqslant|u_{k^2}|+\tfrac2kC. $ When $n\to\infty$, $k\to\infty$ (since $k\gt\sqrt{n}-1$) hence $|u_{k^2}|\to0$ and $\tfrac2k\to0$, which proves that $|u_n|\to0$.