I was working on some integrals and I came across the following series: $$\sum_{k=1}^{\infty}\frac{1}{k(k+n)!}$$ Wolfram Alpha evaluates it to be $$\frac{_2F_2\left(1,1;2,n+2;1\right)}{(n+1)!}$$ Which is a hypergeometric series. I searched on the Internet but I didn't find any useful info and I tried some simple methods to evaluate it but didn't reach anything. I would appreciate any hint, resolution or resource you could give me. I would also appreciate some info about how to evaluate the hypergeometric series shown above.
2026-03-27 00:02:34.1774569754
Infinite sum related to hypergeometric series
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