Product of even numbers divided by product of odd numbers

Number Theory Level 4

Suppose you multiply even numbers from 1 to 10 -> 2 x 4 x 6 x 8 x 10

and divide that product by the product of odd numbers from 1 to 10 -> 1 x 3 x 5 x 7 x 9

You get a finite real number which is less than 5.

Now let us extend this operation to a higher number n, where the numerator is 2 x 4 x 6 x .... x 2n ............................(1)

and the denominator is 1 x 3 x 5 x 7 x ... x (2n-1) ...............(2)

Dividing (1) by (2) gives you the result.

Let us call this Res(n)

Show how you arrive at the approximate formula for Res(n)

Find an approximate value for this result for n = 50 and n = 500

Let P = Res(500)/Res(50)

What is the value of P correct to two decimal places.

The answer is 3.16.

1 solution

Maggie Miller
Aug 28, 2015

We have $Res(n)=\frac{2\cdot 4\cdots 2n}{1\cdot 3\cdots (2n-1)}=\frac{(2\cdot 4\cdots 2n)^2}{1\cdot 2\cdot 3\cdots 2n}=\frac{2^{2n}(n!)^2}{(2n)!}$ .

From Stirling's approximation, $\frac{2^{2n}(n!)^2}{(2n)!}\approx\displaystyle\frac{2^{2n}\left(2\pi n\left(\frac{n}{e}\right)^{2n}\right)}{\sqrt{4\pi n}\left(\frac{2n}{e}\right)^{2n}}=\sqrt{n\pi}$ .

Therefore, $\frac{Res(500)}{Res(50)}\approx\frac{\sqrt{500\pi}}{\sqrt{50\pi}}=\sqrt{10}\approx\boxed{3.16}$ .

Product of even numbers divided by product of odd numbers

The answer is 3.16.

1 solution

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