The GregoryLeibniz series is another method for approximating pi from "summary" of A History of [pi] (pi) by Petr Beckmann
Another method for approximating pi is the Gregory-Leibniz series. This series involves summing an infinite number of terms, with each term becoming smaller and smaller as the series progresses. The formula for the Gregory-Leibniz series is 4/1 - 4/3 + 4/5 - 4/7 + 4/9 -... and so on. By adding up more and more terms, one can get closer and closer to the value of pi. The Gregory-Leibniz series is a straightforward way to estimate pi, but it converges very slowly. This means that it takes a large number of terms to get an accurate approximation of pi. For example, to get just two decimal places of precision using the Gregory-Leibniz series, one would need to sum hundreds of terms. Despite its simplicity, the Gregory-Leibniz series is not the most efficient method for approximating pi. Although the Gregory-Leibniz series is not the most efficient method for calculating pi, it is a valuable tool for understanding the concept of infinite series and the properties of pi. By exploring the Gregory-Leibniz series, one can gain insight into the nature of pi and the relationship between circles and their circumference. Additionally, the Gregory-Leibniz series serves as an introduction to more advanced methods of approximating pi, such as the Machin formula and the Ramanujan series.- While the Gregory-Leibniz series may not be the most efficient method for approximating pi, it is a valuable tool for exploring the concept of pi and infinite series. By studying the Gregory-Leibniz series, one can deepen their understanding of pi and its significance in mathematics and science.
