Convergence of $\left( \frac{1}{1} \right)^2+\left( \frac{1}{2}+\frac{1}{3} \right)^2+\cdots$ Does the following series converge? If yes, what is its value in simplest form?
$$\left( \frac{1}{1} \right)^2+\left( \frac{1}{2}+\frac{1}{3} \right)^2+\left( \frac{1}{4}+\frac{1}{5}+\frac{1}{6} \right)^2+\left( \frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} \right)^2+\dots$$
I have no idea how to start. Any hint would be really appreciated. THANKS!

Since for moderately large values of $n$ we have
$$ H_n \approx \log(n)+\gamma+\frac{1}{2n}-\frac{1}{12n^2} $$
we also have
$$ H_{n(n+1)/2}-H_{n(n-1)/2} \approx \frac{2}{n}-\frac{4}{3n^3}$$
and
$$ \sum_{n\geq 1}\left[H_{n(n+1)/2}-H_{n(n-1)/2}\right]^2 \approx 1+\sum_{n\geq 2}\left(\frac{2}{n}-\frac{4}{3n^3}\right)^2=1+\frac{2 \left(-1890+2835 \pi ^2-252 \pi ^4+8 \pi ^6\right)}{8505} $$
is finite and approximately equal to $\color{green}{3.17}151$. I doubt there is a simple closed form.
