How do you find the sum of factorials $1! + 2! + 3!+ ................... + n!?$ ?

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Answer 1 · 2016-05-23T04:41:51

The formula below computes this sum

$\sum_{k = 0}^{n} k! = \frac{i\pi}{e} + \frac{\text{Ei}(1)}{e} - \frac{(-1)^n\ \Gamma[n+2]\ \Gamma[-n-1, -1]}{e}$

Where $Ei$ is the Exponential Integral function, and $Γ[x]$ is the Euler Gamma Function whilst $Γ[x,n]$ is the upper incomplete Gamma Function.

How do you find the sum of factorials 1! + 2! + 3!+ ................... + n!?1! + 2! + 3!+ ................... + n!??