[SSS-{8}] Sum of the reciprocals of the triangle numbers

I got my secret summer santa question from League. The question is as follows:

Does the sum of the reciprocals of all triangular numbers converge, and if it does, what is its value?

To answer this question we first need to know that the expression for the ith triangle number is i(i+1)/2. Which means that we are summing over 2/(i(i+1)) from i=1 to infinity. We know that it converges because it is strictly less than 2 times the p-series for p=2. Next we evaluate the sum.

By doing partial fraction decomposition we get a telescoping series and from there the sum is quite straightforwardly evaluated.