Q: Limit distribution when no of terms is infinite 
If $\lim _{n \rightarrow \infty} \frac{1^{n}+2^{n}+3^{n}+4^{n}+\dots+n^{n}}{n^{n}}=k,$ then $\left[\frac{10}{k}\right]=\ldots$ (Where [.] denotes G.I.F)

I was able to calculate using Stolz-Cesàro lemma, it came out to be $\frac{e}{e-1}$. But can there be another method for students who are not yet exposed to Stolz-Cesàro lemma?
