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How Madhava of Sangamagrama discovered infinite series, proto-calculus, and convergence acceleration ~250 years before Newton and Leibniz
Around 1350 CE, when Europe was emerging from the Middle Ages and recovering from the Black Death, a mathematician in a Kerala village was performing calculations that would not be matched in Europe for another 250+ years.
Infinite series for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + ... (Madhava-Leibniz series). In Europe only in 1673.
Infinite series for sine: sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + ... (Madhava-Newton series). Credit to Newton ~1670.
Infinite series for cosine: cos(x) = 1 − x²/2! + x⁴/4! − x⁶/6! + ... Credit to Newton.
Pi to 11 decimal places: 3.14159265359 — not matched in Europe until ~1600 CE.
Convergence acceleration: Correction terms for series giving faster convergence — 300+ years before Euler.
Madhava's original works are mostly lost. His work is known through quotations in later Kerala School texts — particularly the Yuktibhasha (Jyeshthadeva, ~1530) and Tantrasangraha (Nilakantha Somayaji, ~1501).
In the 20th century, scholarship (particularly K.V. Sarma's work) clarified the extent of Madhava's contributions. In 1994, mathematician Victor Katz confirmed that Madhava's series are identical to Newton and Leibniz's "Taylor series."