# A question concerning Cesaro-Stolz theorem

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

• the numerator is not $(n+1)^{n+1}$. – Nosrati Aug 8 '18 at 4:42
• @user108128 Thanks, I notice it now. – xbh Aug 8 '18 at 4:44
• But why is the denominator not $(n+1)^{n+1}$? – Szeto Aug 9 '18 at 15:09
• @Szeto It should be $$\sum_1^{n+1} j^{n+1} - \sum_1^n j^n$$, which clearly does not equal $(n+1)^{n+1}$ – xbh Aug 9 '18 at 15:10
• @xbh Ah! I even typed ‘denominator’ in my comment! I better get some rest. – Szeto Aug 9 '18 at 15:14