1/(2^2-1) +1/(3^2-1) +1/(4^2-1) +...+1/(19^2-1) +1/(20^2-1)
= 1/(2^2-1^2) +1/(3^2-1^2) +1/(4^2-1^2) +...+1/(19^2-1^2) +1/(20^2-1^2)
=1/(2+1)(2-1) + 1/(3+1)(3-1) +……+1/(20+1)(20-1)
=1/2*3 + 1/3*4 +...+1/20*21
=(1/2 - 1/3) + (1/3-1/4) + (1/4-1/5)...+(1/19 - 1/20)
=1/2 - 1/20
=9/20