1/1 + 2/(2+3) + 3/(4+5+6) + 4/(7+8+9+10) + ...

It seems to me that the solution is

��(x/(((x^2+x)/2)^2+(x^2+x)-(((x-1)^2+x-1)/2)^2+((x-1)^2+x-1)/2)

from 1 to ��.

Assuming the same range, this simplifies to

��(x/((x^3+x)/2))

or 2*��(2/(x^2+x)

or 2*1, which is 2.