Ramanujan–Sato Infinite Series in C++

Overview

This project implements one of the most famous infinite series discovered by Srinivasa Ramanujan for calculating π.

Published in 1914, Ramanujan's series for ( \frac{1}{\pi} ) is known for its incredible rate of convergence, producing approximately 8 additional correct decimal digits of π per term.

The formula is:

$$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)! (1103 + 26390k)} {(k!)^4 396^{4k}} $$

Because of its extremely fast convergence, even the first term ((k = 0)) produces an approximation of π accurate to several decimal places.

This project explores:

• Implementing arbitrary precision calculations in C++
• Understanding infinite series convergence
• Comparing calculated values of π against the true value
• Measuring runtime and accuracy across multiple iterations

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Features

• Calculates π using Ramanujan's infinite series
• Supports increasing precision with additional terms
• Measures program execution time
• Compares computed π with the actual value of π

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Example

Using only the first term:

Calculated π: 3.141592653589793 Actual π: 3.141592653589793

Adding more terms rapidly increases accuracy due to the exponential convergence of the series.

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Future Improvements

• Add arbitrary precision support using libraries such as Boost.Multiprecision
• Visualize convergence over multiple iterations
• Compare Ramanujan's series against other π algorithms
• Benchmark performance improvements

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References

• S. Ramanujan, "Modular equations and approximations to π", 1914