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Math Help - convergence or divergence of series

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    convergence or divergence of series

    examine the convergence or divergence of (summation) (log(base n) n!)/(n^3)

    can we estimate n! with the help of stirlings formula and then do the question
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    Re: convergence or divergence of series

    Quote Originally Posted by prasum View Post
    examine the convergence or divergence of (summation) (log(base n) n!)/(n^3)

    can we estimate n! with the help of stirlings formula and then do the question
    Claim: \log_{n}(n!) \le n

    You can prove this by induction if needed, but observe

    \log_{n}(n!)=\log_{n}(n)+\log_{n}(n-1)+\log_{n}(n-2)+...\log_{n}(2)+\log_{n}(1) \le \underbrace{1+1+1+...+1+1}_{n \text{ times}}=n

    Or if you belive that n! \le n^n and since the log function is monotonic increaseing you get

    \log_{n}(n!) \le \log_{n}(n^n)=n

    But either way you end up with

    \sum_{n=1}^{\infty}\frac{\log_{n}(n!)}{n^3} \le \sum_{n=1}^{\infty}\frac{n}{n^3}=\sum_{n=1}^{\inft  y}\frac{1}{n^2}

    And we know that this converges.
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