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Stirling's Formula: How does n! relate to n^n.

  1. Nov 17, 2012 #1
    Hello all.
    I am doing a school paper on Stirling's formula and I want first to show how the factor e comes in to place. So I found somewhere on the net the definition n! = (1-1/2)×(1-1/3)2... × (1-1/n)n-1] × nn.

    I will then use this result and the definition of e^x since we are talking for large n to proceed.

    1. The problem statement, all variables and given/known data
    Problem is, how do I prove it? Although I haven't tried induction yet, I was told that I should find an algebraic method.

    I don't want a complete solution, just a hint to guide me through the proof.

    Thank you
     
    Last edited: Nov 17, 2012
  2. jcsd
  3. Nov 17, 2012 #2
    First represent all the 1-.5 and 1-1/3 by (1-1/n+1)^n, we note this is (n/(n+1))^n.
    Now note what happens as we multiply this for different n's:
    (1/2)^1 * (2/3)^2 * (3/4)^3... = (1^1 * 2^2 * 3^3...)/(2^1 * 3^2 * 4^3...).
    Do you notice what is happening? Try canceling things up, in problems like this, I recommend you put things in variables like I did (n/(n+1)) in our case.


    Thanks, Bonaparte
     
    Last edited: Nov 17, 2012
  4. Nov 17, 2012 #3
    Thank you for your response. I got it now, the fraction reduces to n!/(n+1)^n = Product from 1 to n of (n/(n+1))^n which leads to n!=(n+1)^n * the Product, but then the last term of the product cancels out with (n+1)^n and so comes the required result.

    However, I am still interested on how you derived it. I mean, suppose we had absolutely no clue on how does n^n relate to n!. How would we then, think of this relationship? What steps would we have to take?
     
  5. Nov 17, 2012 #4
    The relationship between n^n and n! is ruled by their definition. Now that I think about it a clearer way to have stated this is instead of n/n+1 (where we have to change n^n to (n+1)^n) is to state it in their way (1-1/n)^(n-1).
    Then we have ((n-1)/n)^(n-1). Then proceed as before, although now it will be clearer. Considering your writing a paper about it, I recommend the second way.

    As for your question, I derived it by look at the numbers, seeing the pattern (which is not even necessary considering its given) and implementing it into numbers.

    Thanks and good luck, Bonaparte
     
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