Find if the sequence $n!/10^{6n}$ converges Find if the sequence $a_n$ converges, $a_n=\frac{n!}{10^{6n}}=\frac{1}{\frac{10^6n}{n!}}$.  Now if we take the limit of numerator and denominator as $n \to \infty$, we get $\frac{1}{0}$($\lim\limits_{n\to\infty} \frac{x^n}{n!}=0 $  for any $x$), but division by zero is not defined, so we are stuck, but the answer is $\infty$.  Where am I going wrong?

Hint for an alternative route,
$$\rm \frac{a_{\large n+10^6}}{a_{\large 10^6}}\normalsize =\frac{10^6+1}{10^6}\cdot\frac{10^6+2}{10^6}\cdots\frac{10^6+n}{10^6}>\left(1+\frac{1}{10^6}\right)^n \to \infty.$$
