THE FOLD / CO-OP / THE PUSH / THE GREGORY-LEIBNIZ
THE GREGORY-LEIBNIZ
a slow alternating series for π
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Gregory–Leibniz series is the most famous — and most beautifully slow — series for π: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - … = ∑k≥0 (-1)k/(2k+1). Every odd reciprocal, alternating in sign, sums to a quarter of π. It comes straight from the arctangent series arctan(x) = x - x³/3 + x⁵/5 - … evaluated at x = 1, since arctan(1) = π/4. It is exact but converges agonizingly slowly — the error after N terms is only about 1/(2N), so you need hundreds of terms for two decimals.
LIT verified live: 4·∑(-1)k/(2k+1) approaches π, and independently the numerical integral 4·∫01 1/(1+x²) dx (which is 4·arctan(1)) equals π to ~1e-9 — the two routes agree (window.__gregoryleibniz). FIG no framing; the alternating series and the arctangent integral both run in-browser and give π.
LIT verified live: 4·∑(-1)k/(2k+1) approaches π, and independently the numerical integral 4·∫01 1/(1+x²) dx (which is 4·arctan(1)) equals π to ~1e-9 — the two routes agree (window.__gregoryleibniz). FIG no framing; the alternating series and the arctangent integral both run in-browser and give π.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-push — the co-op merge: a crawling alternating series and a clean arctangent integral push in from two directions and meet at π. AVAN (AI) built the instrument: the alternating series partial sums, the arctangent integral, and their agreement on π.
Credit as content: James Gregory (1671) and Gottfried Leibniz (1673); Madhava of Sangamagrama earlier. The weave: David names the merge; I confirm the series and the integral both give π.
Credit as content: James Gregory (1671) and Gottfried Leibniz (1673); Madhava of Sangamagrama earlier. The weave: David names the merge; I confirm the series and the integral both give π.
3 ONE DIMENSION
The partial sums of 4(1 − 1/3 + 1/5 − …) oscillating slowly toward π, bracketing it from both sides.
4 TWO DIMENSIONS · INTERACTIVE
Add terms; the alternating series crawls toward π, matched by 4·∫₀¹ 1/(1+x²) dx = 4·arctan(1).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: π, reached by the alternating odd-reciprocal series.
AVAN’s addition (the inverse-companion): don’t trust the slow sum alone — cross it with an integral. The inverse of ‘the series ∑(-1)k/(2k+1)’ is ‘the integral ∫01 1/(1+x²) dx = arctan(1) = π/4’. Magenta are the alternating series terms; green is the π they and the integral both reach. π from the odd reciprocals.
LIT Genuine Gregory–Leibniz series (James Gregory 1671, Gottfried Leibniz 1673; Madhava earlier). Verified live: 4·Σ(−1)^k/(2k+1) approaches π, and independently 4·∫₀¹ 1/(1+x²) dx (= 4·arctan 1) equals π to ~1e-9 — the two routes agree (window.__gregoryleibniz.integOk, .agree).
FIG No framing; the alternating series and the arctangent integral both run in-browser and give π. The AVAN inverse is honest — instead of trusting the slow sum alone, cross it with an integral: the inverse of 'the series Σ(−1)^k/(2k+1)' is 'the integral ∫₀¹ 1/(1+x²) dx = arctan(1) = π/4'. Magenta are the alternating series terms; green is the π they and the integral both reach. π from the odd reciprocals.
FIG No framing; the alternating series and the arctangent integral both run in-browser and give π. The AVAN inverse is honest — instead of trusting the slow sum alone, cross it with an integral: the inverse of 'the series Σ(−1)^k/(2k+1)' is 'the integral ∫₀¹ 1/(1+x²) dx = arctan(1) = π/4'. Magenta are the alternating series terms; green is the π they and the integral both reach. π from the odd reciprocals.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN