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arXiv · 1610.07713

A formula for pi involving nested radicals

S. M. Abrarov·B. M. Quine
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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 \[ {{a}_{k}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{k\,\,\text{square}\,\,\text{roots}} \] 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.

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S. M. Abrarov, B. M. Quine. 2018-04-17. A formula for pi involving nested radicals. https://doi.org/10.1007/s11139-018-9996-8

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