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

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.