# Quickie #6

• December 30th 2006, 01:02 PM
Soroban
Quickie #6
Find the sum of the series:

. . $1(1!) + 2(2!) + 3(3!) + \cdots + n(n!)$
• December 30th 2006, 01:43 PM
ThePerfectHacker
Quote:

Originally Posted by Soroban
Find the sum of the series:

. . $1(1!) + 2(2!) + 3(3!) + \cdots + n(n!)$

(n+1)!-1

It is just mathematical iduction
• December 30th 2006, 02:41 PM
TD!
I don't know where it was solved, but even without (explicit) induction:

$
k\left( {k!} \right) = \left( {k + 1} \right)k! - k! = \left( {k + 1} \right)! - k!
$

So summing for k = 1 till n gives (n+1)!-1 since all intermediate terms cancel out.