Easy Proofs of Some Borwein Algorithms for $\pi$

TL;DR

This paper provides simple proofs for two of Borwein's algorithms for calculating pi, using only Gauss's formula and basic algebra, simplifying the understanding of these efficient methods.

Contribution

It introduces elementary algebraic proofs for two Borwein algorithms, making their correctness more accessible without advanced techniques.

Findings

Proved correctness of two Borwein algorithms using elementary algebra.

Demonstrated that Gauss's formula suffices for these proofs.

Simplified the understanding of efficient pi computation algorithms.

Abstract

In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\pi$. We will see that by using only a formula of Gauss's and elementary algebra we are able to prove the correctness of two of them.