Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus

TL;DR

This paper presents a calculus-based method to derive closed-form formulas for sums of arithmetic sequences raised to powers, extending classical results and incorporating Bernoulli numbers through elementary differentiation and integration techniques.

Contribution

It introduces a novel symbolic calculus approach for summing powered arithmetic sequences, including polynomially increasing steps, with natural emergence of Bernoulli numbers.

Findings

Derives closed-form expressions for sums of powered arithmetic sequences.

Extends classical formulas to sequences with polynomial steps.

Links calculus techniques to Bernoulli number-based formulas.

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.