I am trying to see whether or not this series converges for $a_k>0,k\geq 0$ and $\sum_k a_k^2 < \infty$.
When $a_k \geq 1$, I have this bound: $$ \begin{align*} \sum_{k=1}^N \frac{a_k}{\sum_{i=0}^k a_{i}^2} &\leq \sum_{k=1}^N \frac{a_k^2}{\sum_{i=0}^k a_{i}^2}\\ &= \sum_{k=1}^N \int_{\sum_{i=0}^{k-1} a_{i}^2}^{\sum_{i=0}^{k} a_{i}^2}\frac{1}{\sum_{i=0}^{k} a_{i}^2}dx\\ &\leq \sum_{k=1}^N \int_{\sum_{i=0}^{k-1} a_{i}^2}^{\sum_{i=0}^{k} a_{i}^2}\frac{1}{x}dx\\ &= \int_{a_0^2}^{\sum_{i=0}^{N} a_{i}^2}\frac{1}{x}dx \end{align*} $$
Suppose that $\sum\limits_{k=1}^{\infty}a_k^2=S>0$. Denote $$ b_k:=\frac{a_k}{\sum\limits_{i=1}^{k}a_i^2}. $$ Then, $$ b_k\sim\frac{a_k}{S},~\text{when}~n\to\infty. $$ Hence, ($a_k, b_k>0$) the series $\sum\limits_{k=1}^{\infty}b_k$ is convergent if and only if the series $\sum\limits_{k=1}^{\infty}{a_k}$ is convergent. Moreover, note that if $\sum\limits_{k=1}^{\infty}a_k$ is convergent, then so is $\sum\limits_{k=1}^{\infty}a_k^2$.