On the periodic continued radicals of 2 and generalization for Vieta product

TL;DR

This paper investigates the limits of periodic continued radicals of 2, showing they converge to specific sine-based values and their subsequences relate to algebraic multiples of pi, extending Vieta's product for pi.

Contribution

It establishes the convergence of periodic continued radicals of 2 to 2sin(qπ), characterizes their limit points, and generalizes Vieta's product for pi.

Findings

Periodic continued radicals of 2 converge to 2sin(qπ).

Limit points of scaled radicals are algebraic multiples of π.

Results generalize Vieta's product for π.

Abstract

In this paper we study periodic continued radicals of 2. We show that any periodic continued radicals of 2 converges to 2sin(q\pi), for some rational number q depends on the continued radical. Furthermore we show that if r_n is a periodic nested radicals of 2, which has n nested roots, then the limit points of the sequence 2^n(2sin(q\pi)-r_n) have the form \alpha\pi, where \alpha is an algebraic number. This result give a set of sub sequences converges to \alpha\pi, for each \alpha . Also we show that limit of these sub sequences can be represented as Vieta like nested radical products. Hence this result generalizes the Vieta product for \pi. Several interesting examples are illustrated.