Consider:
$$\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$$
This is, as far as I'm able to check with my software, correct to at least 167 decimals.
If anyone has the resources and time to perform more accurate calculations to determine where LHS and RHS start to differ, it would be very much appreciated.
It would be even better if someone could show it analytically.