1. ## Prove this limit

$\lim_{n\to&space;\infty}\prod_{k=1}^n&space;{{n+k}\choose{k}}^{1/n^2}&space;=&space;\frac{4}{\sqrt&space;e}$

2. About the power involved there, I think it covers the whole product.

Where it says "lím" is because I took it from a spanish forum, but it clearly means "limit."

(You can solve the double integral as you please.)

3. Good solution! I don't see any ambiguity about the exponent applying to the whole product or not, though.

4. Hi guys can anyone please explain the fourth line? I don't understand how you get 1/n in the negative part. I keep getting 1/n^2. Many thanks.

5. If you write the second product as $\prod_{j=1}^n\frac jn$ you will get the $\frac 1n$ because it doesn't depend on $i$.

6. When you bring the $\frac{1}{n^2}$ in, wouldn't you get - $\frac{1}{n^2}$ $\sum_{i=1}^n$ln $\frac{i}{n}$ ? How did the $\frac{1}{n}$ vanish?