What is the sum of the reciprocals of all the nonzero
triangular numbers?

I know this works for the sum of reciprocals of powers of two:
x = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
2x = 2 + 1 + 1/2 + 1/4 + 1/8 + ...
2x = 2 + x
x=2

But trying the same thing with the triangular numbers seems to require adding terms in pairs:
x = 1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28 + ...
2x = 2 + 2/3 + 2/6 + 2/10 + 2/15 + 2/21 + 2/28 + ...
2x = 2 + (2/3 + 2/6) + (2/10 + 2/15) + (2/21 + 2/28) + ...
2x = 2 + 1 + 1/3 + 1/6 + ...
2x = 2 + x
x=2

I can't remember when this sort of thing is allowed in series like this, but it seems to give the right answer here.

Posted by Jer on 2015-02-20 10:07:29