Method 3 · 1670s
An infinite sum of fractions
π / 4 = 1 − 1⁄3 + 1⁄5 − 1⁄7 + 1⁄9 − …
Just keep alternating + and − over the odd numbers. Forever. You get π.
The estimate (purple) bounces above and below π (gold dashed line) on every new term. Notice how slowly the swings shrink — even at 10,000 terms you barely have three correct digits.
What you're looking at
Take 1. Subtract one-third. Add one-fifth. Subtract one-seventh. Keep going forever, alternating signs, with denominators climbing through the odd numbers. Multiply the whole running total by 4.
That's it. No circles, no triangles, no radii. Just an infinite alternating sum of unit fractions — and somehow it equals π. The chart above shows the running estimate after each new term: a purple line that overshoots π, undershoots, overshoots a little less, undershoots a little less, and very gradually settles in.
Where on earth does this come from?
It comes from arctangent. There's an identity in calculus that says arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … for small enough x. Plug in x = 1: arctan(1) = 1 − 1/3 + 1/5 − 1/7 + … And arctan(1) = π / 4, because the tangent of 45° is 1. So the right side equals π / 4. Multiply by 4. Done.
It's named after Gottfried Leibniz, who published it in 1676 — but it had already been discovered around 1500 in India by mathematicians of the Kerala school, including Madhava. It's sometimes called the Madhava–Leibniz series for that reason.
It's beautiful and useless
Beautiful, because the formula is so unreasonably simple — alternating odd- denominator fractions, what could be tidier?
Useless, because the convergence is glacial. To get just 10 correct decimal places, you'd need around 10 billion terms. To get 100 correct places, more terms than there are atoms in your laptop. Mathematicians figured out tricks to accelerate it, and other formulas (like Machin's, in 1706) that converge much faster were quickly preferred for actual computation. The Leibniz series is mostly remembered as proof that something deep is going on between odd numbers and circles.
Try this
- Add terms one at a time for the first ten or so, watching the estimate flip above and below π.
- Hit Auto-add. Watch how slowly the swings shrink — orders of magnitude more terms for each extra digit.
- Compare to Nilakantha, where a small tweak makes everything dramatically faster.
Sources & further reading
- Wikipedia: Leibniz formula for π — derivation from the arctangent series.
- Wikipedia: Madhava series — the original Indian discovery, ~1500.
- Wikipedia: Kerala school of astronomy and mathematics — the broader tradition the series came from.