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.

Bernoulli's Inequality says that for $n\ge k$,
$$
\left(1-\frac kn\right)^n
$$
is an increasing sequence. Therefore, by Monotone Convergence
$$
\begin{align}
\sum_{k=0}^n\left(\frac kn\right)^n
&=\sum_{k=0}^n\left(\frac{n-k}n\right)^n\\
&=\sum_{k=0}^n\left(1-\frac kn\right)^n\\
&\to\sum_{k=0}^\infty e^{-k}\\
&=\frac e{e-1}
\end{align}
$$
