Summing a Series Like n*(n!)

Date: 10/28/2001 at 02:34:37
From: Aberlig
Subject: Summing up series like n*(n!)

Hello,

I was recently reading a book and was wondering how to add up a 
series whose nth term is n*(n!) where the ! sign indicates factorial, 
so n! = n*(n-1)* ... 2*1. 

The series went like 1*1! + 2*2! + 3*3! ... n*n!. 

Please also show the steps and the method used to solve this problem.

Thank you,
Aberlig

Date: 10/29/2001 at 08:15:06
From: Doctor Floor
Subject: Re: Summing up series like n*(n!)

Hi, Aberlig,

Thanks for writing.

We will show that 1*1! + 2*2! + ... + n*n! = (n+1)! - 1.

This can be done by mathematical induction.

First we note that 1*1! = 2! - 1, so for n = 1 the formula is correct.

Then suppose that 1*1! + 2*2! + ... + n*n! = (n+1)! - 1. We will show 
that then 1*1! + 2*2! + ... + n*n! + (n+1)*(n+1)! = (n+2)! - 1. 

Let's go:

   1*1! + 2*2! + ... + n*n! + (n+1)*(n+1)! = 
          (n+1)! - 1        + (n+1)*(n+1)! =
   (n+2)*(n+1)! - 1 =
   (n+2)! - 1

And we have proved the formula by mathematical induction.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/