Calculate The Sum of The Following Truncated Madhava Leibniz Series

Introduction

The Madhava-Leibniz series is one of the earliest known series representations of π. It was discovered independently by the Indian mathematician Madhava of Sangamagrama in the 14th century and the German mathematician Gottfried Wilhelm Leibniz in the 17th century.

The series is given by:

This series converges to π/4, meaning that as the number of terms increases, the sum approaches π/4. The calculator allows you to compute the sum of the first n terms of this series.

Formula

The sum of the first n terms of the Madhava-Leibniz series is given by:

Where:

• S is the sum of the series
• n is the number of terms
• k is the term index

This formula alternates between adding and subtracting terms to approximate π/4.

Calculation

To calculate the sum of the truncated Madhava-Leibniz series:

1. Determine the number of terms (n) you want to include in the series.
2. Initialize the sum (S) to 0.
3. For each term from k = 0 to n-1:
   • Calculate the term value: (-1)^k / (2k + 1)
   • Add the term to the sum S
4. The final sum S is the approximation of π/4.

The more terms you include, the more accurate the approximation becomes.

Example

Let's calculate the sum of the first 5 terms of the Madhava-Leibniz series:

Calculating each term:

• Term 1 (k=0): 1
• Term 2 (k=1): -1/3 ≈ -0.3333
• Term 3 (k=2): 1/5 = 0.2
• Term 4 (k=3): -1/7 ≈ -0.1429
• Term 5 (k=4): 1/9 ≈ 0.1111

Summing these terms:

The exact value of π/4 is approximately 0.7854. Our approximation with 5 terms is 0.8355, which is reasonably close.

Interpretation

The sum of the truncated Madhava-Leibniz series provides an approximation of π/4. The accuracy of the approximation depends on the number of terms included:

• With fewer terms, the approximation is less accurate.
• With more terms, the approximation becomes more precise.
• The series converges to π/4 as the number of terms approaches infinity.

This series is historically significant as one of the earliest known series representations of π. It demonstrates the power of infinite series in mathematics and provides a practical way to approximate π.