Sums of Reciprocals of Generalized Triangular Numbers

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

💡 Research Summary

The paper investigates the infinite series formed by the reciprocals of triangular numbers and their higher‑order generalizations. It begins with the classical triangular numbers Tₙ = n(n + 1)/2 and asks for the values of the non‑alternating sum Σₙ≥1 1/Tₙ and the alternating sum Σₙ≥1 (−1)^{n+1}/Tₙ. Two standard calculus techniques are employed: a telescoping series method and a power‑series method.

In the telescoping approach, the identity 1/