设k为大于1的正整数
则1/((k-1)k(k+1))
=k/((k-1)k*k(k+1))
=(k(k+1)-k(k-1))/2((k-1)k*k(k+1))
=(1/(k(k-1))-1/(k(k+1)))/2
所以原式=(1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+……+1/(20*21)-1/(21*22))/2
=(1/(1*2)-1/(21*22))/2
=115/462