1*1! + 2*2! + ... + n*n!

There is a simple and elegant formula for the sum of the series $1 \cdot 1! + 2 \cdot 2! + \dots + n \cdot n!$. The derivation is left to the reader, but the result is very useful:

$\sum_{k=1}^{n} k \cdot k! = (n+1)! - 1$

For example, to compute $1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4!$, we can use the formula with $n=4$:

$(4+1)! - 1 = 5! - 1 = 120 - 1 = 119$.

Variations may involve leaving out terms, which requires subtracting the missing terms from the total sum.