Let

T_{r} = \frac{1}{2} r (r + 1)

denote the rth triangular number. Prove that the sum of the reciprocals of the first n triangular numbers is approximately equal to 2 when n is large, that is:

\sum_{r = 1}^{n} \frac{1}{T_{r}} = \frac{1}{T_{1}} + \frac{1}{T_{2}} + \frac{1}{T_{3}} + . . . + \frac{1}{T_{n}} ≅ 2

Hence show that the sum of the reciprocals of the first n triangular numbers tends to 2 as n tends to infinity.