Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate the limit

Question

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate: $$\lim_{n\to \infty}a^{-nk}\prod ^k_{j=1}\left(a+\frac{j}{n}\right)^n$$

Clueless on this problem. Seek your help.

Answer 1

We have: (a + j/n)^n = a^n*((1 + j/an)^(an))^(1/a). So let S = the product, we have: Limit(S) = (e^(1+2+..+k))^(1/a) = e^(k(k+1)/2a).

Answer 2

Factoring out the $a$ term and simplifying the inner terms in the product gives: $$a^{-nk}\prod_{j=1}^k(a+\frac{j}{n})^n=\prod_{j=1}^k(1+\frac{j}{an})^n$$ $$\lim_{n\to\infty} a^{-nk}\prod_{j=1}^k(a+\frac{j}{n})^n=\prod_{j=1}^k\lim_{n\to\infty}(1+\frac{j}{an})^n$$ $$\lim_{n\to\infty} a^{-nk}\prod_{j=1}^k(a+\frac{j}{n})^n=\prod_{j=1}^ke^{\frac{j}{a}}=e^{\sum_{j=1}^k\frac{j}{a}}=e^{\frac{k(k+1)}{2a}}$$