для n=1 верно , переход k->n+1 тогда нужно доказать что
n(n+1)/(2*(2n+1))+(n+1)^2/((2(n+1)-1)(2(n+1)+1)) = (n+1)(n+2)/(2*(2n+3))
(n(n+1)(2n+3)+ 2*(n+1)^2)/(2*(2n+1)(2n+3)) = (n+1)(n(2n+3)+2(n+1))/(2*(2n+1)(2n+3)) = (n+1)(2n+1)(n+2)/(2*(2n+1)(2n+3)) = (n+1)(n+2)/(2(2n+3))