Without explicit computation prove that
pr is less than 1/10, but more than 1/15.

pr is as defined.

Let ps = 2/3*4/5*6/7.....98/99, clearly a larger number.

then pr*ps=1/100

So (pr)^2<1/100; so pr<1/10.

The second part is harder. From Weisstein on double factorials,

pr = (2n)!/(2^n*n!)^2, ps = (2^(2(n-1)) ((n-1)!)^2)/((2n-1)!)

and pr exceeds 0.6 of ps even for small pr. So 5/3x^2 = 1/100 and x is greater than 0.7, which is more than 1/15.

Edited on February 13, 2015, 12:59 pm

Posted by broll on 2015-02-13 12:28:19