Ramanujan's Infinite Series for 1/π: Genius Without Proof

Hardy later recalled: "They defeated me completely; I had never seen anything in the least like them before." He showed the letter to his colleague J.E. Littlewood, who remarked that the formulas "must be true, because, if they were not true, no one would have had the imagination to invent them."

The Series for $1/\pi$

Among Ramanujan's most spectacular results were his formulas for $1/\pi$. The most famous appeared in his 1914 paper:

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

This formula is breathtaking for several reasons. First, it converges with extraordinary speed: each successive term adds roughly eight decimal digits of accuracy to $\pi$. The first term alone gives:

$\frac{1}{\pi} \approx \frac{2\sqrt{2}}{9801} \times 1103 = 0.31830988618\ldots$

Compare with the true value: $1/\pi = 0.31830988618\ldots$ — correct to 10 decimal places from a single term.

Second, the constants $9801 = 99^2$ and $396 = 4 \times 99$ hint at deep connections to modular arithmetic and elliptic functions that would not be fully understood for decades.

The Mathematical Framework

Ramanujan's series for $1/\pi$ belong to a family now called Ramanujan-type series or Ramanujan-Sato series. They arise from the theory of modular forms and elliptic integrals.

The general form is:

$\frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{(\frac{1}{2})_n \, (s)_n \, (1-s)_n}{(1)_n^3} \, (An + B) \, z^n$

where $(a)_n = a(a+1)(a+2)\cdots(a+n-1)$ is the Pochhammer symbol (rising factorial), and $A$, $B$, $z$, and $s$ are algebraic numbers determined by the theory of singular moduli — special values of the $j$-invariant of elliptic curves at quadratic irrationalities.

The specific constants in Ramanujan's formula — $1103$, $26390$, $396$ — correspond to the singular modulus associated with $n = 58$ in the theory of complex multiplication. The number $58$ is significant because $e^{\pi\sqrt{58}} \approx 396^4 - 104.000000177\ldots$, a near-integer that reflects deep arithmetic structure.

Ramanujan's Methods

How did Ramanujan discover these formulas? This question has fascinated mathematicians for over a century. He left almost no record of his methods. His notebooks — filled with thousands of results — contain formulas but almost never proofs.

Several factors likely contributed to his extraordinary intuition:

• Pattern recognition — Ramanujan had an unparalleled ability to spot numerical patterns and guess general formulas
• Computational skill — He performed enormous hand calculations, computing special function values to many decimal places
• Modular equations — His notebooks show extensive work with modular equations, which provide the algebraic machinery underlying the $\pi$ series
• Aesthetic sense — He seemed guided by a feeling for which formulas were "right," later describing his insights as revelations from the goddess Namagiri

Hardy and Ramanujan at Cambridge

Hardy arranged for Ramanujan to come to Cambridge in 1914. Their collaboration was one of the most remarkable in mathematical history — Hardy, the rigorous analyst who insisted on proof, and Ramanujan, the intuitive genius who seemed to conjure results from thin air.

Hardy later developed a rating system for mathematical talent on a scale of 0 to 100. He gave himself 25, Littlewood 30, David Hilbert 80, and Ramanujan 100.

Their joint work produced groundbreaking results in number theory, including the Hardy-Ramanujan asymptotic formula for the partition function:

$p(n) \sim \frac{1}{4n\sqrt{3}} \exp\!\left(\pi \sqrt{\frac{2n}{3}}\right)$

This formula estimates how many ways a positive integer $n$ can be written as a sum of positive integers — a question that seems elementary but connects to some of the deepest ideas in mathematics.

The Chudnovsky Formula

In 1987, the Chudnovsky brothers generalized Ramanujan's approach to produce an even faster-converging series:

$\frac{1}{\pi} = 12 \sum_{n=0}^{\infty} \frac{(-1)^n \, (6n)! \, (13591409 + 545140134n)}{(3n)! \, (n!)^3 \, 640320^{3n + 3/2}}$

Each term contributes approximately 14 decimal digits. This formula has been used to compute $\pi$ to trillions of digits and remains the basis of most world-record computations. It is, in a precise sense, a descendant of Ramanujan's original insight.

The Modern Legacy

Ramanujan's series for $1/\pi$ were vindicated by the Borwein brothers in the 1980s, who provided the first rigorous proofs using the theory of modular forms and hypergeometric functions. Since then, mathematicians have discovered hundreds of Ramanujan-type formulas, each corresponding to a different singular modulus.

But the mystery of Ramanujan's intuition remains. He worked without the modern machinery of modular forms. He had no access to a research library, no doctoral training, no computer algebra system. He had only his mind, his slate board, and an extraordinary conviction that mathematical truth could be perceived directly.

A Final Note

Ramanujan died in 1920 at the age of 32, leaving behind notebooks containing roughly 3,900 results. A century later, mathematicians are still mining these notebooks, still proving theorems he stated without proof, still finding connections he glimpsed before anyone else.

His infinite series for $1/\pi$ remain among the most beautiful formulas in all of mathematics — monuments to a mind that could see, in the infinite, patterns of breathtaking precision.

"An equation means nothing to me unless it expresses a thought of God." — Srinivasa Ramanujan