# Sum of Sequence of Reciprocals of Triangular Numbers

## 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

$\blacksquare$