Question

If $N=n!(n\in N,n>2)$, then find $\underset{N\to \infty }{lim}\left[\right(lo{g}_{2}N{)}^{-1}+(lo{g}_{3}N{)}^{-1}+\dots +(lo{g}_{n}N{)}^{-1}]$

Answer 1

$\underset{N\to \infty }{lim}[{\log }_{N}2+{\log }_{N}3+\cdots +{\log }_{N}n][\because ({\log }_{a}b{)}^{-1}=lo{g}_{b}a]$

$={\log }_{N\to \infty }{\log }_N(2.3\dots n)$

$=\underset{N\to \infty }{lim}{\log }_N(n!)$
$=\underset{N\to \infty }{lim}{\log }_{N}N$
$=\underset{N\to \infty }{lim}1$

$=1[\because n!=N]....given$