Let
pr=(1/2)*(3/4)*(5/6)... *(99/100)
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(n1)) ((n1)!)^2)/((2n1)!)
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 20150213 12:28:19 