How Can We Use the Infinite Series to Derive the Value of π?

There are hundreds of interesting formulas used to calculate the mathematical constant pi (π) through the use of infinite series. Some of these formulas are famous, while others are the result of innovative tweaking and remaking. This article delves into a personal development of a new series that serves as a remake of the Madhava-Gregory-Leibniz series, one of the oldest and most studied series for π. Additionally, we explore the Leibniz formula for π and other notable series, such as those by Gregory, Ramanujan, and Wallis.

Keywords: infinite series, Madhava-Gregory-Leibniz series, π

Introduction to Infinite Series and π

π, an irrational and transcendental number, has captivated mathematicians for centuries. Infinite series offer a fascinating way to approximate and derive π. Among these are the Leibniz formula, which is derived from the Taylor series expansion of the arctangent function, and other series that converge much more quickly. This article aims to provide a deep dive into these series, focusing on the Leibniz formula and a personal remake of the Madhava-Gregory-Leibniz series.

The Leibniz Formula for π

The Leibniz formula for π is a fundamental example of an infinite series used to approximate π. The formula is:

pi 4 sum_{n0}^{infty} frac{(-1)^n}{2n 1}

The derivation of this formula involves the Taylor series expansion for the arctangent function. The Taylor series for arctangent is:

tan^{-1} x sum_{n0}^{infty} frac{(-1)^n x^{2n 1}}{2n 1} quad text{for } x leq 1

By substituting x 1 into the series, we obtain:

tan^{-1}1 sum_{n0}^{infty} frac{(-1)^n}{2n 1}

Since tan-11 (frac{pi}{4}), we can write:

frac{pi}{4} sum_{n0}^{infty} frac{(-1)^n}{2n 1}

Multiplying both sides by 4, we get the Leibniz formula:

pi 4 sum_{n0}^{infty} frac{(-1)^n}{2n 1}

This series converges slowly, as indicated by its rate of convergence. To achieve a reasonable approximation of π, a large number of terms must be summed.

A Personal Remake of the Madhava-Gregory-Leibniz Series

A personal remake of the Madhava-Gregory-Leibniz series is given by a telescoping infinite series involving double factorials. This remake was inspired by the original series and uses a variable m that can be any odd natural integer as a convergence speed coefficient. The formula is:

(pi 4 sum_{n1}^{infty} frac{(-1)^{n 1} (2n-1)!!}{(2n)!! (2n 1)})

This series is particularly remarkable for its rapid convergence. Interesting versions of this series include:

Version with Double Factorials: (pi 4 sum_{n1}^{infty} frac{(-1)^{n 1} (2n-1)!!}{(2n)!! (2n 1)}) Version with Single Factorials and Double Factorials: (pi 4 sum_{n1}^{infty} frac{(-1)^{n 1} (2n-1)!}{(2n)! (2n 1)}) Telescoping Series: This involves the manipulation of series terms to cancel out intermediate terms, which can speed up convergence.

Exploring Other Series for Calculating π

While the Leibniz formula is insightful, more advanced series can be more efficient. Here are some notable series:

Gregory's Series

(frac{1}{pi} 2 sqrt{2} cdot frac{2 - sqrt{2}^3}{3 cdot 2^3} cdot sum_{n0}^{infty} frac{4n!}{n!^4 (2n - 1) 16^n})

This series converges relatively quickly, often allowing for a good approximation of π with fewer terms.

Ramanujan's Series

(frac{1}{pi} frac{2 sqrt{2}}{9801} sum_{n0}^{infty} frac{(4n)! (1103 - 26390n)}{n!^4 396^{4n}})

Ramanujan's series converges extremely fast, enabling the computation of π to millions of decimal places with just a few terms.

Wallis's Multiplicative Product Series

Wallis's product series is a unique infinite product that also approximates π:

Wallis's Product Series:

(frac{π}{2} prod_{n1}^{infty} frac{(2n)^2}{(2n - 1)(2n 1)})

This series is particularly interesting due to its multiplicative form and can be used to derive π.

Conclusion

Infinite series provide a rich and fascinating method for deriving π, showcasing the beauty of mathematical analysis and series expansion. While the Leibniz formula is simple and illustrative, more advanced series can yield π with greater efficiency. The exploration of these series not only enriches our understanding of mathematical constants but also highlights the creative and evolving nature of mathematics.