Generalized continued fraction expansions for <i>π</i> and <i>e</i>

Kadyrov, Shirali; Mashurov, Farukh

doi:10.1080/09720529.2021.1936897

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Continued Fractions For E1

The Ramanujan Machine1

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2021

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“…1 (dotted line). It converges much faster than (5). There is a clear signal of period 3, consistent with the recurring pattern (1, 1, n) in (6).…”

Section: Continued Fractions For Esupporting

confidence: 73%

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Derangements and Continued Fractions for $e$

Lynch¹

2021

Irish Math. Soc. Bull.

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Several continued fraction expansions for e have been produced by an automated conjecture generator (ACG) called The Ramanujan Machine. Some of these were already known, some have recently been proved and some remain unproven. While an ACG can produce interesting putative results, it gives very limited insight into their significance. In this paper, we derive an elegant continued fraction expansion, equivalent to a result from the Ramanujan Machine, using the sequence of ratios of factorials to subfactorials or derangement numbers.

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“…1 (dotted line). It converges much faster than (5). There is a clear signal of period 3, consistent with the recurring pattern (1, 1, n) in (6).…”

Section: Continued Fractions For Esupporting

confidence: 73%

“…This is indicated in Raayoni, et al (2020) as a "known" result. A proof was presented by Kadyrov and Mashurov (2019). Lu (2019) gave elementary proofs of other generalized continued fraction formulae for e. However, the connection with derangement numbers was not made by any of these authors.…”

Section: The Ramanujan Machinementioning

confidence: 99%

Derangements and Continued Fractions for $e$

Lynch¹

2021

Irish Math. Soc. Bull.

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show abstract

Generalized continued fraction expansions for π and e

Cited by 1 publication

References 2 publications

Derangements and Continued Fractions for $e$

Derangements and Continued Fractions for $e$

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