re

bonsoir et voilà :

on supose que 1/1.3+1/3.5+......+1/(2n-1)(2n+1)=n/2n+1
et on montre que 1/1.3+1/3.5+......+1/(2n-1)(2n+1)+1/(2n+1)(2n+3)=(n+1)/(2n+3)
on a 1/1.3+1/3.5+......+1/(2n-1)(2n+1)+1/(2n+1)(2n+3)
= n/(2n+1) + 1/(2n+1)(2n+3)
=n(2n+3)+1/(2n+1)(2n+3)
on simplifie n(2n+3) = 2n²+3n+1=2n²+n+2n+1=n(2n+1)+2n+1=(n+1)(2n+1)
alors (n+1)(2n+1) /(2n+1)(2n+3)
= (n+1)/(2n+3)