How can I find the limit without using a closed form expression [duplicate]

Solution 1:

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

Solution 2:

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