What is the limit of a difficult combination?

[wisky40]

Homework Statement

show that $$\displaystyle \lim_{n \to \infty} \left[ \left(\begin{matrix} n \\ 0 \end{matrix} \right) \left(\begin{matrix} n \\ 1\end {matrix} \right) ...\left(\begin{matrix} n \\ n \end{matrix} \right ) \right]^\frac{1}{n^2} = e^\frac{1}{2}$$

Homework Equations

Stirling's approximation $$n! \sim \sqrt{2 \pi n} n^n e^{-n}$$

The Attempt at a Solution

Firstly I tried to use smallest term to the n-power because that is # of terms of these combinations. Then, $$\displaystyle \lim_{n \to \infty} ((n^n))^\frac{1}{n^2} =1$$. Secondly I did the same with de largest term, taking into account that middle term of Pascal's triangle $$\displaystyle \sim \frac{n!}{(\frac{n}{2}!)(\frac{n}{2}!)}$$ which gave me 2, so now I know that the limit is between 2 and 1 but I haven't proved that the limit is $$e^\frac{1}{2}$$. I also tried logs and play with these combinations from right to left and in reverse but it did not help much. If anybody knows any generating function or idea that I can apply, it will be greatly appreciated.
Thanks

 

[Dick]

You can write the product of the C(n,k) as (n!)^n/(0!*1!*...*n!)^2. Take the log and apply Stirling's approximation log(n!)~n*log(n)-n. To estimate the sum of k*log(k) from 1 to n, approximate it by the integral of x*log(x) (much the same way you derive a rough version of Stirling's approximation).

 

[wisky40]

thank you.

After your advice everything was really easy. I didn't look elegant, however, it was effective.