How do you simplify the factorial expression $\frac{(2 n - 1)!}{(2 n + 1)!}$ $((2n-1)!)/((2n+1)!)$ ?

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Answer 1 · 2017-02-08T00:15:12

$\frac{(2 n - 1)!}{(2 n + 1)!} = \frac{1}{(2 n + 1) (2 n)}$ $((2n-1)!)/((2n+1)!) = 1/((2n+1)(2n))$

Remember that:

$n! =n(n-1)(n-2)...1$

And so

$(2n+1)! =(2n+1)(2n)(2n-1)(2n-2) ... 1$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=(2n+1)(2n)(2n-1)!$

So we can write:

$((2n-1)!)/((2n+1)!) = ((2n-1)!)/((2n+1)(2n)(2n-1)!)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =1/((2n+1)(2n))$

Example: $n=10$

$LHS = (19!)/(21!) \ \ \ \ \ = 1/420$
$RHS = 1/(21*20) = 1/420$

How do you simplify the factorial expression ((2n-1)!)/((2n+1)!)(2n−1)!(2n+1)!((2n-1)!)/((2n+1)!)?