How to solve the limit $\lim_{k \to \infty} \frac{(2k)!}{2^{2k} (k!)^2}$.

Question

How to solve this limit??

$$\lim_{k \to \infty} \frac{(2k)!}{2^{2k} (k!)^2}$$

It's a limit, not a series

Answer 1

\begin{align} \lim_{k\to\infty} \frac{(2k)!}{2^{2k}\cdot(k!)^2} &=\lim_{k\to\infty} \frac{(2k)!}{2^k \cdot 2^k \cdot k! \cdot k!} \\ &=\lim_{k\to\infty} \frac{(2k)!}{(2^k \cdot k!)^2} \\ &=\lim_{k\to\infty} \frac{(2k)(2k-1)\cdots(2)(1)}{(2k)^2 (2k-2)^2 \cdots (4)^2 (2)^2} \\ &=\lim_{k\to\infty} \frac{(2k-1)(2k-3)\cdots(1)}{(2k)(2k-2)\cdots(2)} \\ &=0 \end{align}

In the last step, you can think of the fraction as the infinite product of fractions less than $1$ ($\frac{1}{2} \times\frac{3}{4} \times \frac{5}{6} \times\frac{7}{8}...$), which will decrease to $0$.

Answer 2

This is the probability of having $n$ heads and $n$ tails in $2n$ times of a fair coin toss. This probability then simply goes to 0. You can also use the Stirling's approximation to attain this.