Sums of Reciprocals of Generalized Triangular Numbers

TL;DR

Addresses sums of reciprocals of generalized triangular numbers $T_k(n)=\binom{n+k-1}{k}$ by developing two parallel methods: a telescoping decomposition and a power-series construction. For each integer $k>1$, it proves $\sum_{n=1}^{\infty} \frac{1}{T_k(n)}=\frac{k}{k-1}$ and $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{T_k(n)}=k2^{k-1}\log 2 - k\sum_{i=1}^{k-1}\frac{2^{k-1-i}}{i}$, with consistent results verified by both approaches. The power-series route introduces constants $C_j$ tied to harmonic numbers $H_j$ and yields a non-telescoping derivation, while the telescoping route yields a direct closed form via a simple lemma. The results extend the classical $k=2$ case and provide exact, closed-form expressions applicable to higher-order triangular numbers, revealing connections to $\log 2$ and harmonic sums.

Abstract

We compute the sum and the alternating sum of the reciprocals of triangular numbers using two methods: a telescoping series approach and a power series approach. We then extend these results to generalized (higher-order) triangular numbers and obtain closed-form expressions for both the non-alternating and alternating series in all orders.

Figures (1)

• Figure 1: The first five triangular numbers.

Theorems & Definitions (23)

• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Definition 5
• Lemma 6