Evaluate the limit of the following expression Let $n$ and $k$ be positive integers. Let $a>0$. Evaluate:
$$\lim_{n\to +\infty}\frac{\left(a+\frac{1}{n}\right)^n\left(a+\frac{2}{n}\right)^n\cdot\ldots\cdot\left(a+\frac{k}{n}\right)^n}{a^{nk}}$$
I tried to apply the concept of "Limit as a sum", by taking log, but I could not do. 
Somebody, please help!
 A: Hint: Divide top and bottom by $a^{nk}$. To divide the top, divide each term by $a^n$. The typical term becomes $\left(1+\frac{i}{an}\right)^n$.
Finally, use a well-known limit.
A: Convert your function to $\left(1+\frac{1}{an}\right)^{n}\left(1+\frac{2}{an}\right)^{n}...\left(1+\frac{k}{an}\right)^{n}$.  In the limit, each of those factors takes on the form $e^{()}$, so the product becomes a sum.  Can you take it from there?
A: $\lim_{n \to \infty} \frac{(a+\frac{1}{n})^n \cdot \dots \cdot ((a+\frac{k}{n})^n}{a^{nk}}=\lim_{n \to \infty} (1+\frac{1}{an})^n \cdot \dots \cdot (1+\frac{k}{an})^n = \\ \lim_{n \to \infty}((1+\frac{1}{an})^{an})^{\frac{1}{a}} \cdot \dots \cdot ((1+\frac{k}{an})^{an})^{\frac{k}{a}} = e^{\frac{1}{a}} \cdot \dots \cdot e^{\frac{k}{a}} = e^{\frac{(k+1)k}{2a}}$ 
A: Hint: $$ e^{\frac{1}{a}}\cdot e^\frac{2}{a}\cdot\ldots\cdot e^{\frac{k}{a}}=\exp\left(\frac{k(k+1)}{2a}\right).$$
A: We divide both top and bottom by $a^{nk}$ to get
$$
\frac{\left(a+\frac{1}{n}\right)^n\left(a+\frac{2}{n}\right)^n\cdot\ldots\cdot\left(a+\frac{k}{n}\right)^n}{a^{nk}} 
= 
\left(1+\frac{1}{an}\right)^n\left(1+\frac{2}{an}\right)^n\cdot\ldots\cdot\left(1+\frac{k}{an}\right)^n.
$$
This can be rearranged to
$$
\left(1+\frac{1/a}{n}\right)^n\left(1+\frac{2/a}{n}\right)^n\cdot\ldots\cdot\left(1+\frac{k/a}{n}\right)^n.
$$
Now, it is well known that
$$
\lim_{n \to \infty} \left( 1 + \frac{z}{n} \right)^n = e^{z},
$$
so taking limits we have
$$
\lim_{n \to \infty} \left(1+\frac{1/a}{n}\right)^n\left(1+\frac{2/a}{n}\right)^n\cdot\ldots\cdot\left(1+\frac{k/a}{n}\right)^n 
= 
e^{1/a}\cdot e^{2/a}\cdot\ldots\cdot e^{k/a} = e^{\frac{k(k+1)}{2a}}.
$$
The last equality uses the fact that
$$
1 + 2 + \ldots + k = \frac{k(k+1)}{2}.
$$
