How to show that n! < (n+1)^n

  • Thread starter brookey86
  • Start date
  • #1
16
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Homework Statement



n! < (n+1)n

I am not looking for a proof, just a way to simplify this equation mathematically.

Homework Equations





The Attempt at a Solution



As far as I can see, I cannot simplify this any further. Is there something I can divide out of both sides, for example?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
zcd
200
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[tex]n! = \prod_i=1^n i[/tex]
[tex](n+1)^n =\prod_i=1^n (n+1)[/tex]

What can you say about the relation between i and n+1?
 
  • #3
zcd
200
0
[tex]n! = \prod_i=1^n i[/tex]
[tex](n+1)^n =\prod_i=1^n (n+1)[/tex]

What can you say about the relation between i and n+1?

EDIT: bad latex
[tex]n! = \prod_{i=1}^n i[/tex]
[tex](n+1)^n =\prod_{i=1}^n (n+1)[/tex]
 
  • #4
16
0
I can see that i will always be less than n+1. But is there a way to compare the two without using the product summation symbol?
 
  • #5
zcd
200
0
You could confirm what you wrote:
[tex](\frac{1}{n+1})( \frac{2}{n+1} ) ...( \frac{n-1}{n+1} )( \frac{n}{n+1} )< 1 \Rightarrow n! < (n+1)^n[/tex]
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
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Since that formula depends upon the postive integer n, you might consider proof by induction.
 

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