Prove that $\displaystyle \sqrt[n^2]{\binom{n+1}1\binom{n+2}2\cdots\binom{n+n}n}\to\fr ac4{\sqrt e}$ as $\displaystyle n\to\infty.$
First off, rewrite that as $\displaystyle \exp \left\{ \frac{1}{n^{2}}\prod\limits_{i=1}^{n}{\binom{n+i}i } \right\}$ and prove that $\displaystyle \frac1{n^2}\prod\limits_{i=1}^{n}{\binom{n+i}i}=\f rac{1}{n^{2}}\sum\limits_{i=1}^{n}{\sum\limits_{j= 1}^{n}{\ln \frac{i+j}{n}}}-\frac{1}{n}\sum\limits_{i=1}^{n}{\ln \frac{i}{n}}.$
It's a beautiful exercise proving this!