Question

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

Answer 1

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.