TITLE: A question concerning Cesaro-Stolz theorem
QUESTION [5 upvotes]: Someone has posted a question about evaluating
$$
A \colon =\lim_{\mathbb N^* \ni n \to \infty }\frac {1^n + 2^n + \cdots + n^n} {n^n},
$$
and I thought I could apply Cesaro-Stolz theorem, because the denominator $n^n \nearrow +\infty$. But if I apply it, then I get
$$
A = \lim_n \frac {(n+1)^{n+1}} {(n+1)^{n+1} - n^n} = \lim_n \frac {\left( 1 + \dfrac 1n\right)^{n+1}} {\left( 1 + \dfrac 1n\right)^{n+1} - \dfrac 1n} = \frac {\mathrm e}{\mathrm e - 0} = 1,
$$
instead of $\mathrm e / (\mathrm e - 1)$. What happened here? 
Possible duplicate from following post:  Understanding $(\frac{1}{n})^n+(\frac{2}{n})^n+...+(\frac{n}{n})^n$ sum
Older post: How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?

REPLY [2 votes]: Thanks to @user108128. This is purely my mistake. Maybe this is a lesson for someone read the post could learn [or not, cause this mistake is the “inferior” type].