I'm sorry that this is not an answer, but it's worthwhile information that might help.
If we apply the Limit Comparison Test to the two series, putting the "harmonic" series below, we have the ratio $\left(\frac{n\,a_n}{a_1+\cdots+a_n}\right)^2$
Now if $a_n=f(n)$ where $f$ is an increasing continuous function, but not one that increases too quickly (as defined below when it matters) then
$\left(\frac{n\,a_n}{a_1+\cdots+a_n}\right)^2<\left(\frac{n\,f(n)}{\int_0^nf(x)\,dx}\right)^2$
And so if $f$ is slow-growing, as defined by $\int_0^nf(x)\,dx>C\,n\,f(n)$, then this ratio is bounded. So the Limit Comparison Test would give the convergence of $\sum\frac{n^2a_n}{(a_1+\cdots+a_n)^2}$.
I've found this problem to be much harder to tackle for quickly growing $a_n$, which is funny, since for these the series $\sum\frac{1}{a_n}$ has "more room" between it and a divergent series. If $a_n$ is all-the-time "quickly growing", then this lends itself to a direct examination of $\sum\frac{n^2a_n}{(a_1+\cdots+a_n)^2}$, where the denominator can be shown to be larger enough than the numerator to give convergence. But I think the real problem with any continued approach like this will be sequences that go back-and-forth between slowly growing and quickly growing.
And of course there is the concern that $a_n$ might not even be increasing.