A formula for pi involving nested radicals



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.


Constant pi Arctangent function Nested radical 

Mathematics Subject Classification




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.


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© 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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