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

1 Answer

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.