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🔢 Ramanujan’s Infinite Series for π – A Genius Ahead of Time

By Sharadhvi Tirakannavar

When we talk about π (pi), most of us think of 3.14159… and maybe the classic formula π = C/d (circumference over diameter). But what if I told you that over a century ago, a self-taught Indian mathematician discovered formulas for π so powerful that they are still used in today’s fastest algorithms for calculating trillions of digits of π?

That mathematician was Srinivasa Ramanujan, a name etched in mathematical history for his mind-bending contributions. Among his countless discoveries, his infinite series for π stands as one of the most remarkable achievements.

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✅ Why Ramanujan’s Series is Special

Before Ramanujan, mathematicians used slow-converging series to approximate π. These took thousands of steps for even a few accurate digits. Ramanujan’s series, however, converged so quickly that just a few terms could give accuracy up to millions of digits!

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🧮 The Magical Formula

Here’s the famous Ramanujan series:

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

Let’s break this down:

• The factorial (!) grows extremely fast, making each term incredibly small after the first few.

• The denominator has 396 raised to the 4k power, which ensures rapid convergence.

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⚡ How Fast Does It Converge?

Most older formulas needed hundreds of terms to get just a few digits of π. Ramanujan’s formula?

• First term → correct to 8 decimal places!

• Second term → accuracy in 16 decimal places!

This means computers can compute π with mind-blowing precision millions of digits long, thanks to Ramanujan’s insight.

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🖥️ Why is it Still Relevant?

Modern algorithms like the Chudnovsky algorithm, which currently powers world-record π computations, are based on Ramanujan’s ideas. Without his formulas, we might not have achieved such precision in cryptography, simulations, and scientific calculations.

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🌟 A Glimpse of Ramanujan’s Genius

What’s fascinating is that Ramanujan derived these without formal Western training. In his own words:

“An equation for me has no meaning, unless it expresses a thought of God.”

His intuition remains unmatched, and his π series is proof that beauty and complexity in math often come from simplicity and vision.

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🔍 Quick Summary:

✔ Ramanujan’s π formula converges super-fast.
✔ Basis for modern algorithms computing trillions of π digits.
✔ Example of pure genius meeting practical use.

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💡 Fun Fact: In 1985, physicist Simon Plouffe used Ramanujan-inspired formulas to compute over 17 million digits of π on a personal computer—a world record at the time!