I was reading this old question and fascinated by the second infinite sum $$\sum_{n=1}^{\infty}\dfrac{1}{\sqrt{n!}}.$$ This is clearly convergent (by comparison or ratio test) and, we can obtain some crude approximations of this using inequalities like $n^4\le n!\le (n!)^2,$ here the first inequality holds for all $n\ge7.$ But, I wonder if we can find the exact value of this series using (familiar) special functions/constants. How would you attack to this series?
2026-03-28 09:33:35.1774690415
Value of $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n!}}$?
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