But the first few values are:
1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28...
I really don't see any pattern there showing why it converges to 2 exactly
Edit:
After thinking some more, you could write the sum as:
(Sum from n=1 to infinity of): 2/(n * (n + 1))
That sum is smaller than the sum of:
2 * (1/n^2^) which converges to π^2^/3
So I can see why it converges, just not where to.
What about a couple thousand decaffeinated espressos (there is certainly 0.x mg of caffeine in one, not 0)?