Ratio Test for Series

Question

Is there an example for ratio test for series that we need to use L'Hopital rule to find $$ \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$$ because the limit is not a straightforward limit?

Answer 1

Perhaps the series $$\sum_{n=0}^\infty \frac{n!}{n^n}\hspace{5mm}?$$ Then $$\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}}\cdot \frac{n^n}{n!} = \frac{1}{\left(1+\frac{1}{n} \right)^n}.$$ Now, the limit of the right hand side as $n\to \infty$ could certainly involve l'Hopital if you haven't seen it before, but another might immediately recognize the limit as the reciprocal of $e$. So it depends on what you mean by need.