My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
May 13th, 2009, 05:33 AM   #1
Newbie
 
Joined: May 2009

Posts: 25
Thanks: 0

Factorial expressed as the sum of a series - new ?

I noted this result when I was a child
If you take the series 1^n , 2^n, 3^n ........

Then repeatedly take the differences, after n operations you end up with n!

I.e for n=2 then
1,4,9,16,25
3,5,7,9
2,2,2

n=3
1,8.27.64,125, 216
7,19,37,61,91
12,18,24,30
6,6,6

From this by considering the general series

1^n, 2^n, 3^n,........r^n...and taking differences

I derived the general result obtaining the alternating series

n! = sum (r = 0 to n) (-1)^r * nCr * (n+1-r)^n
Where C is the combinatorial function nCr = n!/r!(n-r)!

or sum(r= 0 to n) (-1)^r * (1/(r!(n-r)!) * (n+1-r)^n = 1 for any n

i.e 5! = 6^5 - 5 * 5^5 + 10 * 4^5 - 10 * 3^5 + 5 * 2^5 -1 = 7776 - 15625 + 10240 - 2430 + 160 -1 = 120

Is this result known about elsewhere or am I just showing something trivial like n! = n! ?
CarlPierce is offline  
 
May 13th, 2009, 08:50 AM   #2
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 932

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Factorial expressed as the sum of a series - new ?

You've discovered an important result:



. . .
CRGreathouse is offline  
May 13th, 2009, 09:46 AM   #3
Senior Member
 
Joined: Nov 2007

Posts: 258
Thanks: 0

Re: Factorial expressed as the sum of a series - new ?

That last one is nice. Here it is in LaTeX (more readable)

brunojo is offline  
May 14th, 2009, 01:36 AM   #4
duz
Senior Member
 
Joined: Oct 2008

Posts: 215
Thanks: 0

Re: Factorial expressed as the sum of a series - new ?

Quote:
Originally Posted by brunojoyal
That last one is nice. Here it is in LaTeX (more readable)

We could have

Given n+x boxes and n balls, how much different ways there're to put the n balls into the n+x boxes and none of the first n boxes is empty?
Using Inclusion-Exclusion Principle, the result is the leftside of the equation. And it is obvious the result is n! too.
duz is offline  
May 14th, 2009, 02:29 AM   #5
Newbie
 
Joined: May 2009

Posts: 25
Thanks: 0

Re: Factorial expressed as the sum of a series - new ?

How can I find out if this series result has been obtained before, as seems fairly likely ?

And more importantly why would we expect an alternating set of powers of n+1 -> 1 raised to the n to produce n!
CarlPierce is offline  
October 31st, 2009, 02:29 AM   #6
Newbie
 
Joined: Oct 2009

Posts: 1
Thanks: 0

Re: Factorial expressed as the sum of a series - new ?

Quote:
Originally Posted by duz
Quote:
Originally Posted by brunojoyal
That last one is nice. Here it is in LaTeX (more readable)

We could have

Given n+x boxes and n balls, how much different ways there're to put the n balls into the n+x boxes and none of the first n boxes is empty?
Using Inclusion-Exclusion Principle, the result is the leftside of the equation. And it is obvious the result is n! too.
For this is intuitive and clear. But it works also for .

This is, for me, less clear...
antityzer is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
expressed, factorial, series, sum



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Evaluating a series with factorial nothingisimpossible Calculus 1 August 11th, 2013 03:21 PM
Why can a function be expressed as a Taylor Series? Issler Real Analysis 1 March 14th, 2012 12:56 PM
Sum of Infinite Series With Factorial Red5 Calculus 1 February 27th, 2010 09:12 PM
Factorial prime series karel747 Number Theory 10 January 9th, 2008 10:09 AM
Evaluating a series with factorial nothingisimpossible Algebra 1 January 1st, 1970 12:00 AM





Copyright © 2017 My Math Forum. All rights reserved.