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Math Help - Sequences and series

  1. #1
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    Sequences and series

    If K is positive integer of the series, find the radius of the convergence
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  2. #2
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    \frac{(n!)^k}{(kn)!}
    Ratio test,
    \frac{[(n+1)!]^k}{(kn+k)!}\cdot \frac{(kn)!}{(n!)^k}
    Cancel a lot of stuff,
    \frac{(n+1)^k}{(kn+k)(kn+k-1)...(kn+1)}

    Now the numerator is a polynomial of degree k with leading coefficient 1.

    The denominator is a polynomial of degree k and leading coefficient k^k.

    So the limit of this ratio is \frac{1}{k^k}.

    Which means the radius of convergence is k^k.
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