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How do I begin to evaluate this limit: $$\lim_{n\to \infty}\ \frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}\;?$$

Thanks a lot.

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What limits do you already know? This product can be written as $\prod_{k=1}^n (1-\frac{1}{2k})$. –  Thomas Andrews May 1 '12 at 17:29
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up vote 15 down vote accepted

A simple, but famous trick works here: Observe that $(n-1)(n+1) = n^2 - 1 \leq n^2$. Thus we have $$ \begin{align*} & \left[ \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \right]^2 \\ &= \frac{1 \cdot 1}{2 \cdot 2}\cdot\frac{3 \cdot 3}{4 \cdot 4}\cdot\frac{5 \cdot 5}{6 \cdot 6}\cdots\frac{(2n-3) \cdot (2n-3)}{(2n-2) \cdot (2n-2)}\cdot\frac{(2n-1) \cdot (2n-1)}{(2n) \cdot (2n)} \\ &= \frac{1 \cdot 3}{2 \cdot 2}\cdot\frac{3 \cdot 5}{4 \cdot 4}\cdot\frac{5 \cdot 7}{6 \cdot 6}\cdots\frac{(2n-3) \cdot (2n-1)}{(2n-2) \cdot (2n-2)}\cdot\left(\frac{2n-1}{(2n)^2}\right) \\ & \leq \frac{2n-1}{(2n)^2} \\ & \leq \frac{1}{2n}. \end{align*} $$ Thus we have $$\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \leq \frac{1}{\sqrt{2n}} $$ and the limit is zero. In fact, the sharp estimate $$ \frac{1}{\sqrt{(\pi + o(1)) n}} \leq \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \leq \frac{1}{\sqrt{\pi n}} $$ holds, so the estimation above is not so far from the truth.

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+1. Nice....... –  user17762 May 1 '12 at 17:34
    
You seem to be great when it comes to integrals. Could you help me with this? –  Pedro Tamaroff May 1 '12 at 17:38
    
where did you use the fact that $(n-1)(n+1) = n^2 - 1 \leq n^2$ and why does $\begin{align*} & \frac{1 \cdot 3}{2 \cdot 2}\cdot\frac{3 \cdot 5}{4 \cdot 4}\cdot\frac{5 \cdot 7}{6 \cdot 6}\cdots\frac{(2n-3) \cdot (2n-1)}{(2n-2) \cdot (2n-2)}\cdot\left(\frac{2n-1}{(2n)^2}\right) \leq \frac{2n-1}{(2n)^2} \end{align*}$? –  Anonymous May 1 '12 at 17:54
    
@Anonymous: Note that $$ \frac{(n-1)(n+1)}{n^2} \leq 1.$$ This fact is applied fraction-wise (but for the last term) to the LHS of the inequality. –  sos440 May 1 '12 at 17:58
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@Anonymous: I believe you are understanding the inequality $$0 < \frac{(x-1)(x+1)}{x^2} < 1$$ if $x > 1$. This follows directly from the inequality $0 < (x+1)(x-1) = x^2 - 1 < x^2$. Then plugging $x = 2, 4, \cdots 2n-2$, we have $$ 0 < \frac{1 \cdot 3}{2 \cdot 2} < 1, \quad 0 < \frac{3 \cdot 5}{4 \cdot 4} < 1, \quad \cdots, \quad 0 < \frac{(2n-3)\cdot(2n-1)}{(2n-2)\cdot(2n-2)} < 1.$$ Now multiply these inequalities term-by-term. Then we obtain the desired inequality. –  sos440 May 1 '12 at 18:26
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This is almost a quite famous limit.

Wallis showed that

$$\lim_{n \to \infty} \left(\frac{(2n)!!}{(2n-1)!!}\right)^2 \frac 1 n = \pi$$

or that

$$ \frac{(2n)!!}{(2n-1)!!} \sim \sqrt{\pi n}$$

Your limit is

$$\lim_{n \to \infty} \frac{(2n-1)!!}{(2n)!!}$$

which by the above asymtotical behaviour is

$$\lim_{n \to \infty} \frac{(2n-1)!!}{(2n)!!} = \lim_{n \to \infty} \frac{1}{\sqrt{\pi n}}=0$$

Note you can write you expression as

$$\frac{{\left( {2n - 1} \right)!!}}{{\left( {2n} \right)!!}} = \frac{{\left( {2n - 1} \right)!!}}{{\left( {2n} \right)!!}}\frac{{\left( {2n} \right)!!}}{{\left( {2n} \right)!!}} = \frac{{\left( {2n} \right)!}}{{{4^n}n{!^2}}}$$

so using Stirling's approximation, one has

$$\eqalign{ & \left( {2n} \right)! \sim {\left( {\frac{{2n}}{e}} \right)^{2n}}2\sqrt {n\pi } \cr & n{!^2} \sim {\left( {\frac{n}{e}} \right)^{2n}}2n\pi \cr} $$

from where

$$\frac{{\left( {2n} \right)!}}{{{4^n}n{!^2}}} \sim \frac{1}{{{4^n}}}\frac{{{{\left( {\frac{{2n}}{e}} \right)}^{2n}}2\sqrt {n\pi } }}{{{{\left( {\frac{n}{e}} \right)}^{2n}}2n\pi }} = \frac{1}{{\sqrt {n\pi } }}$$

as it was previously stated.

In general,

$${\left[ {\frac{{\left( {2n} \right)!!}}{{\left( {2n - 1} \right)!!}}} \right]^2}\left( {1 - \frac{1}{{2n + 1}}} \right)\frac{1}{n} < \pi < {\left[ {\frac{{\left( {2n} \right)!!}}{{\left( {2n - 1} \right)!!}}} \right]^2}\frac{1}{n}$$

$$\left[ {\frac{{\left( {2n} \right)!!}}{{\left( {2n - 1} \right)!!}}} \right]\sqrt {1 - \frac{1}{{2n + 1}}} < \sqrt {n\pi } < \left[ {\frac{{\left( {2n} \right)!!}}{{\left( {2n - 1} \right)!!}}} \right]$$

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