Sum of Sequence of Reciprocals of Triangular Numbers
Theorem
- $\ds \sum_{k \mathop \ge 1} \dfrac 1 {T_k} = 2$
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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$