Preprint Article Version 2 This version is not peer-reviewed

On 'Best' Rational Approximations to $\pi$ and $\pi+e$

Version 1 : Received: 15 May 2020 / Approved: 16 May 2020 / Online: 16 May 2020 (16:32:49 CEST)
Version 2 : Received: 31 May 2020 / Approved: 31 May 2020 / Online: 31 May 2020 (16:21:52 CEST)

How to cite: Teguia Tabuguia, B. On 'Best' Rational Approximations to $\pi$ and $\pi+e$. Preprints 2020, 2020050268 (doi: 10.20944/preprints202005.0268.v2). Teguia Tabuguia, B. On 'Best' Rational Approximations to $\pi$ and $\pi+e$. Preprints 2020, 2020050268 (doi: 10.20944/preprints202005.0268.v2).

Abstract

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

Subject Areas

$\pi$ (read Pi); $e$ (the natural logarithm base); continued fraction; convergent infinite series

Comments (1)

Comment 1
Received: 31 May 2020
Commenter: Bertrand Teguia
Commenter's Conflict of Interests: Author
Comment: Changes in the title; some clarity added concerning the aim of the paper, especially the advantage of the used approach: see the emphasis on continued fractions in the abstract, the introduction last sentence, the third section last sentence, and the discussion about the structure of $\pi+e$ in the last section.
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