标题： 通过微积分推导算术序列和的整数次幂的闭式表达式 显示英文标题

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

摘要： 本文介绍了一种基于符号微积分的方法，用于推导等差数列求和的闭式表达式。 该方法不仅适用于常数差数列，还适用于具有多项式增长步长的数列，包括线性、二次、三次及更高阶的形式。 通过使用微分和积分的基本技巧，该方法生成表示总和的多项式表达式，即使每个项都被提升到正整数次幂时也是如此。 因此，伯努利数自然地出现在公式中，以简洁且易于理解的方式将该方法与经典结果联系起来。 显示英文摘要

摘要： 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.