What is the sum of all the factorials starting from 1 to n? Is there any generalized formula for such summation?
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ankit
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1http://oeis.org/A003422 has some information about the related sum which starts from 0 instead of 1. There is almost certainly not a nice formula (unless you would consider $\sum_1^n k!$ to already be a nice formula). – Micah Sep 22 '16 at 04:44
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See also http://oeis.org/A007489. – vadim123 Sep 22 '16 at 04:52
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See this. – Mhenni Benghorbal Sep 25 '16 at 15:27
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This is one of the cases where appears the subfactorial function $$\sum_{k=1}^{n}k!=-1-!1-(-1)^n(n+1)!\times !(-2-n)$$ and,as you will notice in the Wikipedia page, $$!m = \left[ \frac{m!}{e} \right] = \left\lfloor\frac{m!}{e}+\frac{1}{2}\right\rfloor, \quad m\geq 1$$ or, more generally $$!m=\frac{\Gamma (m+1,-1)}{e}$$ where appears the incomplete gamma function (see here).
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Claude Leibovici
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+1. Nice answer. Note a 'minor typo': Wi$\color{#f00}{\large\mathrm{l}}$ipedia. – Felix Marin Sep 23 '16 at 04:40