利用微积分推导整数次幂等差数列和的闭式表达式
Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus
AI总结：
本文提出基于符号微积分的方法，可推导步长多项式增长、项取正整数次幂的等差数列和的闭式表达式，自然引入伯努利数关联经典结论。
摘要：
本文提出了一种基于符号微积分的方法，用于推导等差数列和的闭式表达式。该方法不仅适用于公差恒定的序列，还可推广至步长呈多项式增长的序列，包括线性、二次、三次及更高阶形式。该方法利用微分与积分的基础技巧，即便每一项取正整数次幂，也能生成表示总和的多项式表达式。最终，伯努利数会自然出现在公式中，以简洁易懂的方式将该方法与经典结论建立关联。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
