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The Ramanujan Journal

, Volume 46, Issue 3, pp 657–665 | Cite as

A formula for pi involving nested radicals

  • S. M. Abrarov
  • B. M. Quine
Article
  • 206 Downloads

Abstract

We present a new formula for pi involving nested radicals with rapid convergence. This formula is based on the arctangent function identity with argument \(x=\sqrt{2-{{a}_{k-1}}}/{{a}_{k}}\), where
$$\begin{aligned} {{a}_{k}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{k\,\,\text {square}\,\,\text {roots}} \end{aligned}$$
is a nested radical consisting of k square roots. The computational test we performed reveals that the proposed formula for pi provides a significant improvement in accuracy as the integer k increases.

Keywords

Constant pi Arctangent function Nested radical 

Mathematics Subject Classification

11Y60 

Notes

Acknowledgements

This work is supported by National Research Council Canada, Thoth Technology Inc. and York University. The authors thank the reviewers for constructive comments and recommendations.

References

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    Kreminski, R.: \(\pi \) to thousands of digits from Vieta’s formula. Math. Mag. 81(3), 201–207 (2008)CrossRefzbMATHGoogle Scholar
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    Abrarov, S.M., Quine, B.M.: Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi (2016). arXiv:1604.03752
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    Abrarov, S.M., Quine, B.M.: A simple identity for derivatives of the arctangent function (2016). arXiv:1605.02843

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dept. Earth and Space Science and EngineeringYork UniversityTorontoCanada
  2. 2.Dept. Physics and AstronomyYork UniversityTorontoCanada

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