1/1*3=0.5*(1-1/3)
1/3*5=0.5*(1/3-1/5) ...
1/(2n-1)(2n+1)=0.5*(1/(2n-1)-1/(2n+1))
所以
1/1*3+1/3*5+1/5*7+....+1/(2n-1)(2n+1)=0.5*(1-1/3+1/3-1/5....-1/(2n+1))
=0.5*(1-1/(2n+1))
=n/(2n+1)
1/1*3=0.5*(1-1/3)
1/3*5=0.5*(1/3-1/5) ...
1/(2n-1)(2n+1)=0.5*(1/(2n-1)-1/(2n+1))
所以
1/1*3+1/3*5+1/5*7+....+1/(2n-1)(2n+1)=0.5*(1-1/3+1/3-1/5....-1/(2n+1))
=0.5*(1-1/(2n+1))
=n/(2n+1)