New proofs of Borwein-type algorithms for Pi

TL;DR

This paper presents new, concise proofs of Borwein-type algorithms for calculating Pi, utilizing translation and WZ-methods without relying on modular theory or Gauss' formula.

Contribution

It introduces simplified, self-contained proofs of Borwein algorithms using translation and WZ-methods, avoiding complex modular theory.

Findings

Provides new proofs for quadratic and quartic Borwein algorithms

Offers initial values leading to 1/π limit without modular theory

Proofs are short, self-contained, and accessible

Abstract

We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit $1/\pi$. We will not use the modular theory nor either the Gauss' formula that we used in another paper. Our proofs are short and self-contained.