I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this?
Like there is a generic formula for the series:
$$ 1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}{2} $$
or
$$ 1^{2} + 2^{2} + 3^{2} + 4^{2} + \cdots + n^{2} = \frac{n(n+1)(2n + 1)}{6} $$
Is there is any formula for:
$$ 1! +2! +3! + 4! + \cdots + n! $$
and
$$ {1!}^2 +{2!}^2 +{3!}^2 + \cdots + {n!}^2 $$?
Thanks in advance.
If not, is there any research on making this type of formula? Because I am interested.