Loading...
Loading...
Madhava"s infinite series for π, sine, cosine, and arctangent — with correction terms, worked examples, and the story of the world"s first calculus textbook
Between 1350 and 1550 CE, in a small village in Kerala, a lineage of mathematicians achieved what seemed impossible. Without printing presses, without universities, without international communication \u2014 they discovered infinite series, computed \u03C0 to 11 decimal places, and laid the foundations of calculus. All of this happened 250\u2013340 years before Newton and Leibniz.
π/4 = 1 − 1/3 + 1/5 − 1/7 + ...
This means: add 1, subtract 1/3, add 1/5, subtract 1/7... and continue forever. But the raw series is painfully slow \u2014 even after 1,000 terms, you only get 2 correct decimals.
Madhava's genius was inventing correction terms. By adding a correction factor after summing N terms \u2014 just 50 terms yield \u03C0 accurate to 11 decimal places. Europe developed comparable techniques ~400 years later.
sin(x) = x − x³/3! + x⁵/5! − ...
cos(x) = 1 − x²/2! + x⁴/4! − ...
In the West, these are called "Taylor/Maclaurin series" (Taylor 1715, Maclaurin 1742). Madhava derived them ~1400 CE \u2014 300+ years earlier.
Verification: sin(30\u00B0) = sin(\u03C0/6): Term 1 = 0.5236, Term 2 = \u22120.0239, Term 3 = +0.0003 \u2192 Sum = 0.5000 \u2714 (five-decimal accuracy from just 3 terms!)
The arctan(x) = x \u2212 x\u00B3/3 + x\u2075/5 \u2212 ... series with x = 1 gives \u03C0/4, but converges slowly. Madhava chose x = 1/\u221A3, giving \u03C0/6 = arctan(1/\u221A3) \u2014 where each term shrinks much faster. This clever substitution enabled high precision from fewer terms.