Question: Find $\lim_{n \to \infty} \left[ \frac{(m+1) (m+2) \cdots (m+n)}{n^n} \right]^{\frac{1}{n}}$.
My attempt:
Let $L = \lim_{n \to \infty} \left[ \frac{(m+1) (m+2) \cdots (m+n)}{n^n} \right]^ {\frac{1}{n}} $ = $\lim_{n \to \infty} \left[(\frac{m}{n} + \frac{1}{n})(\frac{m}{n} + \frac{2}{n})(\frac{m}{n} + \frac{3}{n})\cdots(\frac{m}{n} + \frac{n}{n})\right]^{\frac{1}{n}},$
$\ln(L)$ = $\lim_{n \to \infty}\frac{1}{n} \sum_{k=1}^{n} \ln(\frac{m}{n} + \frac{k}{n}).$
How can I proceed further from here?
