convergence or divergence of series

**prasum** · September 25th 2012, 11:36 AM

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

**TheEmptySet** · September 25th 2012, 12:09 PM

Originally Posted by prasum

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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