Sums of Reciprocals of Generalized Triangular Numbers

TL;DR

This paper calculates the sums and alternating sums of reciprocals of triangular and generalized triangular numbers using calculus, providing closed-form formulas for all cases and extending classical results to higher-order cases.

Contribution

It introduces new closed-form expressions for sums of reciprocals of generalized triangular numbers, extending classical results to higher orders.

Findings

Closed-form formulas for sums of reciprocals of triangular numbers

Extension of results to generalized higher-order triangular numbers

Derivation of formulas for both non-alternating and alternating series

Abstract

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.