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

  1. Dec 7, 2011 #1
    1. The problem statement, all variables and given/known data

    n! < (n+1)n

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

    2. Relevant equations



    3. 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?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 7, 2011 #2

    zcd

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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?
     
  4. Dec 7, 2011 #3

    zcd

    User Avatar

    EDIT: bad latex
    [tex]n! = \prod_{i=1}^n i[/tex]
    [tex](n+1)^n =\prod_{i=1}^n (n+1)[/tex]
     
  5. Dec 7, 2011 #4
    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?
     
  6. Dec 7, 2011 #5

    zcd

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    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]
     
  7. Dec 7, 2011 #6

    HallsofIvy

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    Science Advisor

    Since that formula depends upon the postive integer n, you might consider proof by induction.
     
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