The Bauer–Ramanujan formula: historical analyses and perspectives
Bibliographic record
Abstract
The formula 2π=1−5⋅(12)3+9⋅(1⋅32⋅4)3−⋯was famously included as a discovery in Ramanujan's first letter to Hardy in 1913, and has been referred to as the Bauer–Ramanujan formula, in view of Bauer's 1859 proof of the above formula. There is a rich history associated with this formula and its many and dramatically different proofs, including a computer-based proof due to Zeilberger that may be seen as groundbreaking in the history of computer-assisted proofs. In addition to a complete survey we provide of all known proofs of the Bauer–Ramanujan formula, we introduce historical analyses based on these proofs, by arguing that the history of the Bauer–Ramanujan formula and our account of this history may be seen as being representative of much broader trends in the history of mathematics. In this regard, the earlier proofs tend to rely on one of the oldest and most basic tools in classical analysis, namely, interchanging the order of limiting operations. In contrast, the more modern proofs tend to rely on computer-related approaches toward summation problems, as in with Zeilberger-type and Gosper-type telescoping arguments.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".