How do I evaluate this $ \lim \frac{\zeta(n)} {({n)!}} , n\to\infty $ if it was existed?

Question

Is there someone who can show me how do I evaluate this limit

$$ \lim_{n\to +\infty} \frac{\zeta(n)} {n!} $$ if it exists ?

Thank you for any help.

Answer 1

It is trivially zero, because $\zeta$ is a positive decreasing function over the integers $n\geq 2$.

Answer 2

First note that $$ 0<\zeta(n)<\zeta(2) $$ for $n>2$. Thus $$ 0\leq\lim_{n\to\infty}\left|\frac{\zeta(n)}{n!}\right|<\lim_{n\to\infty}\frac{\zeta(2)}{n!}. $$ Can you finish from here?