Sum of Sequence of Reciprocals of Triangular Numbers

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Theorem

$\displaystyle \sum_{k \mathop \ge 1} \dfrac 1 {T_k} = 2$

where $T_k$ denotes the $k$th triangular number.


Proof

\(\ds \sum_{k \mathop \ge 1} \dfrac 1 {T_k}\) \(=\) \(\ds \sum_{k \mathop \ge 1} \dfrac 2 {k \paren {k + 1} }\) Closed Form for Triangular Numbers
\(\ds \) \(=\) \(\ds 2 \sum_{k \mathop \ge 1} \dfrac 1 {k \paren {k + 1} }\)
\(\ds \) \(=\) \(\ds 2 \times 1\) Corollary to Sum of Sequence of Products of Consecutive Reciprocals

$\blacksquare$


Sources