Method 4 · ~1500

A much faster series

π = 3 + ⁴⁄_2·3·4 − ⁴⁄_4·5·6 + ⁴⁄_6·7·8 − …

A small structural tweak compared to Leibniz, and suddenly you don't need the lifetime of the universe to get a few digits.

What just happened

Both series above are doing the same kind of thing: alternating positive and negative fractions, summed forever, with the right limit being π. But the structure is different. Leibniz has denominators that grow linearly: 1, 3, 5, 7. Nilakantha has denominators that grow as a product of three consecutive numbers: 2·3·4, 4·5·6, 6·7·8.

That cubic growth in the denominator means each term is much, much smaller than the last. Smaller terms means smaller swings around π. Smaller swings mean faster convergence. The result is dramatic: where Leibniz needs around 10 billion terms for 10 correct digits, Nilakantha needs about 10,000.

Who was Nilakantha?

Nilakantha Somayaji was an Indian mathematician and astronomer working in Kerala in the late 1400s and early 1500s. He published this series in his work Tantrasamgraha around 1501 — over 170 years before Leibniz, and as part of a broader tradition called the Kerala school that had also discovered the Madhava–Leibniz arctangent series and other infinite series for π and trigonometric functions.

For a long time, Western histories of math credited the European development of infinite series with the discovery of these formulas. Modern scholarship has steadily restored credit to the Kerala mathematicians, who were doing this work centuries earlier without the formal calculus that Newton and Leibniz would later build.

A pattern, not a one-off

Nilakantha's series isn't the end of the story — it's an early example of a giant pattern. Once mathematicians realized you could speed up convergence by changing the structure of the terms, the floodgates opened. Machin (1706) used arctangent identities to get a series that converged even faster. Ramanujan (early 1900s) found series that gain 8 digits per term. The Chudnovsky brothers (1980s) found one that gains roughly 14 digits per term — and it's still the algorithm modern π record-setters use today.

That whole arc — from polygons to series to Ramanujan to the Chudnovskys — is the story of how raw convergence speed has driven π record-setting for centuries. Nilakantha is where the chase for faster convergence began.

Try this

• Hit Auto-add. Watch how quickly Nilakantha (teal) flattens onto π while Leibniz (purple) is still wildly bouncing.
• Toggle "Zoom in on Nilakantha" to see its tiny remaining wobbles around π — orders of magnitude smaller than Leibniz's.
• Compare matching digits between the two estimates after 100 terms. (It's not even close.)

Sources & further reading

• Wikipedia: Nilakantha Somayaji — biography and mathematical work.
• Wikipedia: Madhava series — the broader Kerala tradition of π series.
• Wikipedia: Tantrasamgraha — the 1501 work where Nilakantha published this series.
• Wikipedia: Approximations of π — places Nilakantha in the convergence-rate arms race.