TITLE: Explicit value for $\sum_{n=1}^{\infty} \left(\frac{1}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}\right)$
QUESTION [12 upvotes]: This question came out from this other one:
Is there an explicit value for this series?
$$\sum_{n=1}^{\infty}\frac{1}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}= \sum_{n=1}^{\infty} \frac{1}{\sum_{i=1}^{n}\sqrt{i}}=\sum_{n=1}^{\infty}\frac{1}{H_{n,-\frac{1}{2}}}$$
As someone pointed out in comments to the other question, this value is a little bit lower than $3.167830$ and it's not far away from $\frac{63}{20}+\frac{7\sqrt{3}}{680}$.
Can we go further on this? Can we reach an exact value or a better approximation?

REPLY [3 votes]: Just computing, the best value I obtained is $$\sum_{n=1}^{\infty}\frac{1}{H_{n,-\frac{1}{2}}}=3.16782938772351480$$ which can be rationalized as $$\frac{5908}{1865}\qquad \frac{7720}{2437}\qquad \frac{15893}{5017}\qquad\frac{87638}{27665}\qquad \frac{247021}{77978}\qquad\frac{828701}{261599}\qquad\frac{3067783}{968418}\qquad \frac{6964267}{2198435}$$ with increasing accuracy. The decimal representation of the last one is $3.16782938772353970$.
Inverse symbolic calculators did not find anything very close to this number.