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

Take the result that we want Ó2/(ii+i).  This is 2Ó1/(ii+i) and is 2Ó((1/i)-(1/(1+i)).  Write this out:

2 times ((1/1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...)

Rearrange parentheses:

2 times (1/1+(0)+(0)+(0)+...)

So 2 times 1 is 2.