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Math Help - ratio test problem

  1. #1
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    ratio test problem

    sum n=1 oo n^n / (n)!

    a_n+1 / a_n =

    edit: looked funny when i typed it in...
    (n+1)^(n+1) / (n+1)! times n! / n^n

    =

    [n^n(n+1)^n(n+1)] / n+1 = n^n(n+1)^n

    did i do the algebra right? what do i do next? ><
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  2. #2
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    Quote Originally Posted by jeph View Post
    sum n=1 oo n^n / (n)!
    Ratio test.

    (n+1)^(n+1)/(n+1)! * n!/n^n

    (n+1)^(n+1)/[(n+1)*n!] * n!/n^n

    (n+1)^n/n^n

    [(n+1)/n]^n

    [1+1/n]^n

    The limit is e>1

    Thus it does not converge.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Ratio test.
    [(n+1)/n]^n

    [1+1/n]^n

    The limit is e>1

    Thus it does not converge.
    how did you turn the first n to a 1 in the first part?

    how did you get e? i dont get it...
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  4. #4
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    Quote Originally Posted by jeph View Post
    how did you turn the first n to a 1 in the first part?

    how did you get e? i dont get it...
    (n+1)/n = (n/n)+(1/n)=1+1/n

    Now the limit you need to memorize is that,

    (1+1/n)^n ---> e
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