MHB Solving Limit Problem: n→∞, n!-1/n³ln(n!)

[Lisa91]

Could anyone tell me please why the limit of this guy is infinity?

\lim_{n\to\infty} \frac{n!-1}{n^{3} \ln(n!)}

 

[Plato2]

Could anyone tell me please why the limit of this guy is infinit
\lim_{n\to\infty} \frac{n!-1}{n^{3} \ln(n!)}

See what you can do with this inequality.

$\ln \left( {n!} \right) = \sum\limits_{k = 1}^n {\ln (k)} \leqslant \sum\limits_{k = 1}^n k =\frac{{n(n + 1)}}{2}$

 

[chisigma]

The 'core' of the problem is to demonstrate that...

$\displaystyle \lim_{n \rightarrow \infty} \frac{n!}{n^{m}} = \infty$ (1)

... for all integers m>0. That is easily achieved supposing n>m, writing ...

$\displaystyle \frac{n!}{n^{m}}= \frac{n\ (n-1)\ (n-2)\ ...\ (n-m+1)}{n^{m}}\ (n-m)\ (n-m-1)\ ...\ 2 = $

$\displaystyle = 1\ (1- \frac{1}{n})\ (1-\frac{2}{n})\ ... (1-\frac{m-1}{n})\ (n-m)\ (n-m-1)\ ...\ 2$ (1)

... and observing what happens if n tends to infinity...

Kind regards$\chi$ $\sigma$