Proof (product of fractions)

[Jafongo]

I need to prove that the product of 1/2 * 3/4 * 5/6 . . . 99/100 is less than 1/10. I understand that the product will continue to approach zero at a decreasing rate as the series progresses, but I need a mathematical explanation (of the most simple terms possible) as to why the product is less than 1/10.

Thank you very much, this problem has been driving me crazy.

 

[galactus]

\(\displaystyle \prod_{k=1}^{50}\frac{2k-1}{2k}<\frac{1}{10}\)

I have been pondering this for a little while and I think that perhaps Stirling's formula may come into play.

Stirling's formula is \(\displaystyle n!\approx n^{n}e^{-n}\sqrt{2n{\pi}}\)...This should be 'asymptotic to', not approximately. I could not get the tilde sign to work.

The product of the even numbers is \(\displaystyle 50!\cdot 2^{50}\). ( I knew this from fooling around with stirling a while back).

Which leads to \(\displaystyle \frac{100!}{(50!)^{2}\cdot 2^{100}}\)

Now, if we use stirling, it whittles down to \(\displaystyle \sqrt{\frac{1}{50{\pi}}}=.079788...\).

I think this is on the right track. See if you can find something more definitive.