Application of a new iterative formula for computing $\pi$ and nested radicals with roots of $2$

Abstract

In this work, we obtain an iterative formula that can be used for computing digits of $\pi$ and nested radicals of kind $c_n/\sqrt{2-c_{n-1}}$, where $c_0=0$ and $c_n=\sqrt{2+c_{n-1}}$. We also show how with the help of this iterative formula, the two-term Machin-like formulas for $\pi$ can be generated and approximated. Some examples with Mathematica codes are presented.

arXiv comment36 pages

References in corpus (5)

• A formula for pi involving nested radicals
• Three essays on Machin's type formulas
• A generalized series expansion of the arctangent function based on the enhanced midpoint integration
• An iterative method for computing $\pi$ by argument reduction of the tangent function
• A rational approximation of the two-term Machin-like formula for $\pi$