i would like to find a formula that is true for $n \geq 1$ for that : $\displaystyle \sum _{k=1}^n \frac{n^k}{k! \times k}$ I already tried many things like telescoping the sum or symetrising it but i cant find anything at all.. It makes me think about $e^x$ series but its not really it and n doesn't go to infinity so it doesn't works. I take any hint !
2026-03-27 21:33:59.1774647239
Compute this numerical series
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Maple says $$ \sum _{k=1}^{n}{\frac {{n}^{k}}{k!\,k}}=n\; {\mbox{$_2$F$_2$}(1,1;\,2,2;\,n)}-{\frac {{n}^{n+1}\; {\mbox{$_2$F$_2$}(1,n+1;\,n+2,n+2;\,n)}}{\Gamma \left( n+2 \right) \left( n+1 \right) }} $$ You will probably not find anything simpler.