Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus
Abstract
This paper introduces a symbolic calculus-based approach for deriving closed-form expressions for the sums of arithmetic sequences. The method extends beyond constant-difference sequences to those with polynomially increasing steps, including linear, quadratic, cubic, and higher-order forms. Using elementary techniques from differentiation and integration, the approach produces polynomial expressions that represent total sums, even when each term is raised to a positive integer power. As a result, Bernoulli numbers emerge naturally in the formulas, linking the approach to classical results in a concise and accessible manner.
Keywords
Cite
@article{arxiv.2507.13402,
title = {Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus},
author = {Ahmed Abdalmuhsin Abdalsahib},
journal= {arXiv preprint arXiv:2507.13402},
year = {2025}
}
Comments
The mathematical derivation requires significant deepening as key justifications are currently insufficient. While results remain valid, the presentation lacks necessary rigor. This version is withdrawn for major revision. A mathematically complete replacement will be submitted separately