# Factorial quotient with same number of factors

I have this limit:

$$\lim_{x\to \infty} \frac{n!(3n)!}{((2n)!)^2}$$

How can I resolve it? I know that it results $$+\infty$$

• it must be $n\to \infty$ – farruhota May 23 at 6:17

Note that $$\frac{(3n)!}{(2n)!} = \frac{(3n)(3n-1)(3n-2)...}{(2n)(2n-1)(2n-2)...} = (3n)(3n-1)(3n-2)...(2n+2)(2n+1)$$ and similarly $$\frac{(2n)!}{n!} = \frac{(2n)(2n-1)(2n-2)...}{(n)(n-1)(n-2)...} = (2n)(2n-1)(2n-2)...(n+2)(n+1).$$ So $$\frac{n!(3n)!}{(2n)!(2n)!} = \frac{(3n)(3n-1)(3n-2)...(2n+2)(2n+1) }{(2n)(2n-1)(2n-2)...(n+2)(n+1)}\ge \left(\frac{3}{2}\right)^n,$$ which is unbounded.

• Thank you! This is the answer I'm looking for – rik99 May 22 at 21:56

You can use Stirling's approximation $$n! \approx n^ne^{-n}\sqrt{2 \pi n}$$. Plug that in and see what happens to the numeric factors. All the $$n$$s will disappear, as will the $$e$$s

• How can I use it for (2n)! and (3n)! ? – rik99 May 22 at 21:43
• Plug them in. $(2n)! \approx (2n)^{(2n)}e^{-2n}\sqrt{2\pi (2n)}$ – Ross Millikan May 22 at 21:50

HINT

One way is to cancel the terms: $$\frac{n! (3n)!}{(2n)!^2} = \frac{\prod_{k=1}^n k \times \prod_{k=1}^{3n} k} {\prod_{k=1}^{2n} k^2} = \frac{(2n+1) \times \ldots \times 3n} {(n+1) \times \ldots \times 2n} = \prod_{k=1}^n \frac{2n+k}{n+k} = \prod_{k=1}^n \left(1 + \frac{n}{n+k}\right)$$ so each product term is actually above one, and later ones are closer to $$1.5$$, and the number of such terms increases without bound as $$n \to \infty$$.

As an inductive step, for $$n\ge1$$, \begin{align} \frac{n!(3n)!}{(2n)!^2} &=\frac{n\,3n(3n-1)(3n-2)}{2n(2n-1)\,2n(2n-1)}\frac{(n-1)!(3n-3)!}{(2n-2)!^2}\\ &=\frac{27}{16}\frac{\left(1-\frac1{3n}\right)\left(1-\frac2{3n}\right)}{\left(1-\frac1{2n}\right)\left(1-\frac1{2n}\right)}\frac{(n-1)!(3n-3)!}{(2n-2)!^2}\\ &=\frac{27}{16}\frac{1-\frac1n+\frac2{9n^2}}{1-\frac1n+\frac1{4n^2}}\frac{(n-1)!(3n-3)!}{(2n-2)!^2}\\ &\ge\frac32\frac{(n-1)!(3n-3)!}{(2n-2)!^2} \end{align}

Interesting is the case of $$A_n=\frac{(a n)!\, (b n)!}{\left(\left(\frac{a+b}{2} n \right)!\right)^2}$$ Using three times Stirling approximation $$\log(A_n)= \left(a \log (a)+b \log (b)-(a+b) \log \left(\frac{a+b}{2}\right)\right) n+\log \left(\frac{2 \sqrt{a b}}{a+b}\right)+O\left(\frac{1}{n}\right)$$