A series specifying factorial

Question

Is there any mathematical contraction of the following factorial series function?

$$ F(n) = 1.1! + 2.2! + 3.3! + ...... + n.n! $$

I tried it by inspecting that $$ x.x! = x^2.(x-1)!$$

I need derivation of the series.

Answer 1

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

Answer 2

Hint: Notice that $$n.n!=(n+1)!-n!$$