Evaluating the limit : $ \lim_{n \to \infty} \frac{ \sum_{k=1}^n n^k}{ \sum_{k=1}^n k^n}$ Here I'm given this limit.
$$\displaystyle \lim_{n \to \infty} \dfrac{\displaystyle \sum_{k=1}^n n^k}{\displaystyle \sum_{k=1}^n k^n}$$
$\displaystyle \sum_{k=1}^n n^k$ simplifies to $\dfrac{n(n^n-1)}{n-1}$ but I'm unable to tackle $\displaystyle \sum_{k=1}^n k^n$.
How do you evaluate this limit?

Note that 
$$ \sum_{k=0}^n k^n = \sum_{j=0}^n (n-j)^n = n^n \sum_{j=0}^n (1-j/n)^n$$
and using dominated convergence, 
$$ \sum_{j=0}^n (1-j/n)^n \to \sum_{j=0}^\infty e^{-j} = \frac{e}{e-1}$$
Thus 
$$ \frac{\sum_{k=0}^n n^k}{\sum_{k=0}^n k^n} \to \frac{e-1}{e}$$
