π convergent denominators grow exponentially at rate e^β
For almost all real numbers, ln(qₙ) grows linearly at slope β ≈ 1.1865. The denominators of π's convergents (1,7,106,113,33102…) grow faster on average due to the anomalous partial quotient 292.
Growth rates of convergent denominators compared
Comparison of denominator growth rates for golden ratio versus typical number
| φ = [1;1,1,1,…] | Bilangan tipikal |
|---|---|
| qₙ wächst wie φⁿ ≈ 1,618ⁿ | qₙ wächst wie (e^β)ⁿ ≈ 3,276ⁿ |
| Langsamstmögliches Wachstum | Lévys Satz |
Continued fraction convergents of π: denominator growth
The partial quotient 292 at step 5 makes π's denominators grow much faster than average. For a "typical" number the ratio ln(qₙ)/n → β ≈ 1.187.
| n | Hasil bagi parsial aₙ | Konvergen pₙ/qₙ | Penyebut qₙ | ln(qₙ)/n |
|---|---|---|---|---|
| 1 | 3 | 3/1 | 1 | 0,00 |
| 2 | 7 | 22/7 | 7 | 0,97 |
| 3 | 15 | 333/106 | 106 | 1,55 |
| 4 | 1 | 355/113 | 113 | 1,19 |
| 5 | 292 | 103993/33102 | 33102 | 2,52 |
| 6 | 1 | 104348/33215 | 33215 | 1,74 |
| 7 | 1 | 208341/66317 | 66317 | 1,54 |
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