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Thread: 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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    $\displaystyle \frac{(n!)^k}{(kn)!}$
    Ratio test,
    $\displaystyle \frac{[(n+1)!]^k}{(kn+k)!}\cdot \frac{(kn)!}{(n!)^k}$
    Cancel a lot of stuff,
    $\displaystyle \frac{(n+1)^k}{(kn+k)(kn+k-1)...(kn+1)}$

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

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

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

    Which means the radius of convergence is $\displaystyle k^k$.
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