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Let $T_r = \frac{1}{2} r(r+1) $denote the $r$th 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} \cong 2 $

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

Summation of series. Mathematical induction. Generalising. Limits. Continued fractions. Triangle numbers. Geometric sequence. Calculus generally. Mathematical reasoning & proof. Making and proving conjectures.