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# 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)

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.

## Keywords

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

## Subject

MATHEMATICS & COMPUTER SCIENCE, General Mathematics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Commenter: Bertrand Teguia

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