A formula for pi involving 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.

10 pages

Cited by in corpus (6)

• A generalized series expansion of the arctangent function based on the enhanced midpoint integration
• Algorithmic determination of a large integer in the two-term Machin-like formula for pi
• An iterative method for computing $\pi$ by argument reduction of the tangent function
• A rational approximation of the two-term Machin-like formula for $\pi$
• Application of a new iterative formula for computing $\pi$ and nested radicals with roots of $2$
• A method to reduce the Lehmer measure in a multi-term Machin-like formula for $\pi$