How can I find the limit without using a closed form expression I am trying to evaluate this limit without using the closed form expression for the sum of natural numbers raised to $k$th power.  $$\lim_{n \to \infty} \dfrac{ 1^n +2^n+\cdots +n^n}{n^n}$$ 
So far I have tried l'Hôpital which complicates it rather than simplifying and Cesaro Stolz doesn't seem to work either.

$$\lim_{n \to \infty} \dfrac{ 1^n +2^n+\cdots +n^n}{n^n} = \lim_{n \to \infty}\frac{n^n}{n^n}+\frac{(n-1)^n}{n^n}+\frac{(n-2)^n}{n^n}+\cdots$$
$$=\lim_{n \to \infty}1+(1-1/n)^n+(1-2/n)^n +\cdots=1+e^{-1}+e^{-2}+\cdots$$ 
then one can sum the geometric series. 
