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Series question

  1. Nov 23, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]\Sigma[/tex]n!/n^n

    index n=1 to infinity


    2. Relevant equations



    3. The attempt at a solution
    Using the Ratio test (limit as n goes to infinity of a[tex]_{n+1}[/tex]/a[tex]_{n}[/tex])
    and found that the series converges.

    However, I thought that factorials grew faster than exponential functions. Therefore, it would diverge, right?

    Could someone explain why? Did I just do something wrong?
     
  2. jcsd
  3. Nov 23, 2008 #2

    ptr

    User Avatar

    Exponents of the form n^n grows much faster than factorials of the form (n!) because the factorial is a multiplication of n terms, the majority of which are less than n, and the power is a multiplication of n terms, all of which are equal to n.
     
  4. Nov 23, 2008 #3

    Mark44

    Staff: Mentor

    Factorials don't grow faster than exponentials of the sort you're working with. Just think about it: n! = 1 * 2 * 3 * ... * n. You have n factors, of which the largest is n.
    n^n = n * n * n * ... * n. Here you have n factors, all of which are n. Clearly this exponental is larger than the factorial above.
     
  5. Nov 23, 2008 #4

    Avodyne

    User Avatar
    Science Advisor

    Exponentials with a fixed base, like e^n or 2^n, grow more slowly than n!.
     
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