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Math Help - Find value of k for which the series converges.

  1. #1
    Super Member General's Avatar
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    Find value of k for which the series converges.

    Hello

    Problem:
    For which positive integer k is the following series convergent?

    \sum_{n=1}^{\infty} \frac{(n!)^2}{(kn)!}

    The Ratio Test seems a good choice.
    After applying the Ratio Test, I will face the following limit:

    \lim_{n\to\infty} \frac{(n+1)^2(kn)!}{(kn+k)!}

    Then ?
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  2. #2
    MHF Contributor Calculus26's Avatar
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    (kn+k)! = (kn+k)...................(kn+1)(kn)!


    you have (n+1)^2/[(kn+k)...................(kn+1)

    for k = 2

    lim(n+1)^2/ [2(n+1)(2n+1)] = 1/4 and we have convergence

    For k > 2 limit is 0 as denominator is of order 3 or higher

    for k= 1 lim(n+1)^2/(n+1) = infinity
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