What Is the Radius of Convergence in the Sequence Space <em>Seq</em>(R) ?
Bibliographic record
Abstract
Classical real analysis rigorously defines convergence via ε − N criteria, yet it frequently regards the specific entry index N as a mere artifact of proof rather than an intrinsic property. This paper fills this quantitative void by developing a radius of convergence framework for the sequence space Seq(R). We define an index-based radius ρa(ε) alongside a rescaled geometric radius ρ∗a (ε); the latter maps the unbounded index domain to a finite interval, establishing a structural analogy with spatial radii familiar in analytic function theory. We systematically analyze these radii within a seven-block partition of the sequence space, linking them to liminf-limsup profiles and establishing their stability under algebraic operations like sums, products, and finite modifications. The framework’s practical power is illustrated through explicit asymptotic inversions for sequences such as Fibonacci ratios, prime number distributions, and factorial growth. By transforming the speed of convergence into a geometric descriptor, this approach bridges the gap between asymptotic limit theory and constructive analysis, offering a unified, fine-grained measure for both convergent and divergent behaviors.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.041 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.010 |
| Scholarly communication | 0.006 | 0.012 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".