Srinivasa Ramanujan · Mathematics
Ramanujan’s astonishing rapidly converging series for 1/π, each term adding roughly eight correct digits; his mastery of continued fractions and the Rogers-Ramanujan identities; and his audacious, intuitive handling of infinite series that Hardy said no one else could have imagined.
Many of the deepest numbers in mathematics, π among them, can be written as an infinite series: an endless sum of terms that creeps toward the true value. But series differ enormously in how fast they close in. The famous series π/4 = 1 - 1/3 + 1/5 - 1/7 + ... is beautiful but agonisingly slow; to get even ten correct digits of π from it you would need billions of terms. A series is rapidly converging if each term you add corrects many more digits, so that a handful of terms suffices for stupefying accuracy. Ramanujan had an unearthly instinct for constructing such series. In a single paper of 1914, Modular Equations and Approximations to π, he wrote down series for 1/π of a speed no one had seen before, drawing on the deep theory of modular functions that he seemed to carry in his head. Where the classical series inched forward, Ramanujan’s sprinted.
Ramanujan’s most celebrated formula from the 1914 paper expresses 1/π as a constant, 2 times the square root of 2, divided by 9801, multiplied by an infinite sum whose terms are built from factorials and from the number 1103 plus 26390 times the counter. The details are formidable, but the behaviour is the point: with just the first term, taking the sum to stop at zero, the formula already pins π to about eight decimal places, and every additional term you include buys roughly eight more. It looks like a magician’s flourish - where do 9801 and 26390 come from? - and Ramanujan gave no proof. The strange constants are in fact values of deep objects from the theory of modular equations, which is how Ramanujan generated the series. It took mathematicians until 1987, with the work of Jonathan and Peter Borwein, to prove this and Ramanujan’s related π series rigorously and to explain the machinery he had used by instinct.
π was only one theatre for Ramanujan’s genius with infinite processes. He was perhaps the greatest master ever of the continued fraction, an expression built by endlessly dividing, a number plus a fraction whose denominator contains another number plus a fraction, and so on forever. It was his continued-fraction formulae, above all, that had ‘defeated’ Hardy in the 1913 letter. He also rediscovered and deepened the Rogers-Ramanujan identities, two exquisite statements equating certain infinite sums with infinite products, first found by L. J. Rogers in 1894 and unearthed again by Ramanujan around 1917; they turn out to encode surprising facts about partitions and reach into physics. Across all of it runs the same signature: results that arrive without warning, that look impossible, and that turn out to be true. Hardy’s verdict on the strangest of them became the most quoted line about Ramanujan: they must be true, because no one would have had the imagination to invent them.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Ramanujan found infinite series for 1/π of extraordinary speed, worked wonders with continued fractions and the Rogers-Ramanujan identities, and treated even divergent series with a bold intuition. Explain what makes a series ‘rapidly converging’ and why his π series matter.
Leads to Archimedes.
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