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?