| _version_ | 1866901054807867392 |
|---|---|
| author | Lin, Lien |
| author_facet | Lin, Lien |
| contents | <p>We introduce a novel recursive sequence known as Lin’s Harmonic Recurrence Formula, defined by:</p> <p> a₁ = 1 <br> a_n = 1/n + ∑_{k=1}^{n−1} [a_k / (n + k)] for n ≥ 2</p> <p>This sequence blends harmonic elements with memory-based recursion. Each term is constructed from a base harmonic term 1/n and a dynamically weighted average of all preceding terms. Despite its dependence on all past values, the sequence exhibits monotonic decrease and converges numerically to approximately 0.2982.</p> <p>We explore its analytical behavior, prove convergence via monotonicity and boundedness, and discuss its connections to harmonic series, expectation models, and weighted memory systems. Potential applications include combinatorial enumeration, probabilistic modeling, and dynamic system simulations.</p> <p>This work demonstrates how a simple recursive formula can yield rich and subtle mathematical behavior through harmonic influence and structure.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15713563 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Convergence of Lin's Harmonic Recurrence Formula: A New Recursive Sequence Study Lin, Lien harmonic recurrence convergence dynamic memory nonlinear sequence Lin's formula <p>We introduce a novel recursive sequence known as Lin’s Harmonic Recurrence Formula, defined by:</p> <p> a₁ = 1 <br> a_n = 1/n + ∑_{k=1}^{n−1} [a_k / (n + k)] for n ≥ 2</p> <p>This sequence blends harmonic elements with memory-based recursion. Each term is constructed from a base harmonic term 1/n and a dynamically weighted average of all preceding terms. Despite its dependence on all past values, the sequence exhibits monotonic decrease and converges numerically to approximately 0.2982.</p> <p>We explore its analytical behavior, prove convergence via monotonicity and boundedness, and discuss its connections to harmonic series, expectation models, and weighted memory systems. Potential applications include combinatorial enumeration, probabilistic modeling, and dynamic system simulations.</p> <p>This work demonstrates how a simple recursive formula can yield rich and subtle mathematical behavior through harmonic influence and structure.</p> |
| title | Convergence of Lin's Harmonic Recurrence Formula: A New Recursive Sequence Study |
| topic | harmonic recurrence convergence dynamic memory nonlinear sequence Lin's formula |
| url | https://doi.org/10.5281/zenodo.15713563 |