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Math Help - Prove that the sequence is increasing

  1. #1
    MHF Contributor alexmahone's Avatar
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    Prove that the sequence is increasing

    Prove that the sequence a_n=\frac{n^n}{n!} is increasing.
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  2. #2
    Super Member girdav's Avatar
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    Re: Prove that the sequence is increasing

    Compute \frac{a_{n+1}}{a_n}.
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  3. #3
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    Re: Prove that the sequence is increasing

    Quote Originally Posted by alexmahone View Post
    Prove that the sequence a_n=\frac{n^n}{n!} is increasing.
    Can you show \frac{{a_{n + 1} }}{{a_n }} = \left( {1 + \frac{1}{n}} \right)^n ~?
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Prove that the sequence is increasing

    a_{n+1}=\frac{(n+1)^{n+1}}{(n+1)!}=\frac{(n+1)^n}{  n!}>\frac{n^n}{n!}=a_n

    So, a_n is increasing.
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