Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914161617797120 |
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| author | Abdalsahib, Ahmed Abdalmuhsin |
| author_facet | Abdalsahib, Ahmed Abdalmuhsin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus Abdalsahib, Ahmed Abdalmuhsin General Mathematics 11B83 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. |
| title | Deriving Closed-Form Expressions for Arithmetic Sequence Sums Raised to Integer Powers via Calculus |
| topic | General Mathematics 11B83 |
| url | https://arxiv.org/abs/2507.13402 |