Let

Tr =  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:

n ∑ r=1   1 Tr =  1 T1 +  1 T2 +  1 T3 + ... +  1 Tr ≅ 2

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