A formula for pi involving nested radicals

TL;DR

This paper introduces a new rapidly converging formula for pi using nested radicals and arctangent identities, enhancing computational accuracy with increasing nested radical depth.

Contribution

The paper presents a novel pi formula based on nested radicals and arctangent identities, offering improved convergence and accuracy over existing methods.

Findings

The formula converges rapidly as the number of nested radicals increases.

Computational tests show significant accuracy improvements.

The approach provides a new perspective on pi approximation using nested radicals.

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.