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/ |
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