Let

${\displaystyle T_r=\frac{1}{2}r(r+1)}$

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:

${\displaystyle \sum _{r=1}^n\frac{1}{T_r}=\frac{1}{T_1}+\frac{1}{T_2}+\frac{1}{T_3}+...+\frac{1}{T_r}\cong 2}$

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