Evaluating the limit : $ \lim_{n \to \infty} \frac{ \sum_{k=1}^n n^k}{ \sum_{k=1}^n k^n}$

Question

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?

Answer 1

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 $$ \frac{\sum_{k=0}^n n^k}{\sum_{k=0}^n k^n} \to \frac{e-1}{e}$$