THE FOLD / CO-OP / SPLIT SCREEN / THE SOPHOMORE'S DREAM
THE SOPHOMORE'S DREAM
an integral equal to a self-power series
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The sophomore’s dream is a pair of astonishing identities discovered by Johann Bernoulli in 1697, where a function raised to itself integrates to an infinite series over nn: ∫01 xx dx = ∑n≥1 (-1)n-1/nn = 1 - 1/4 + 1/27 - … ≈ 0.7834, and ∫01 x-x dx = ∑n≥1 1/nn = 1 + 1/4 + 1/27 + … ≈ 1.2913. The name teases that the result looks like a naive ‘dream’ a student might wish were true — yet it really is. The trick is to expand xx = ex ln x as a power series and integrate term by term.
LIT verified live: the numerical integrals of xx and x-x over [0, 1] match their respective series ∑(-1)n-1/nn and ∑1/nn to ~1e-6 (window.__sophomore). FIG no framing; the numerical integrals and the self-power series are computed by different routes in-browser and agree.
LIT verified live: the numerical integrals of xx and x-x over [0, 1] match their respective series ∑(-1)n-1/nn and ∑1/nn to ~1e-6 (window.__sophomore). FIG no framing; the numerical integrals and the self-power series are computed by different routes in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — two panels: a smooth self-power integral on one, an nn series on the other, landing on the same dreamlike value. AVAN (AI) built the instrument: the numerical integrals of xx and x-x, the nn series, and their agreement.
Credit as content: Johann Bernoulli (1697). The weave: David names the split screen; I confirm ∫x±x equals the self-power series.
Credit as content: Johann Bernoulli (1697). The weave: David names the split screen; I confirm ∫x±x equals the self-power series.
3 ONE DIMENSION
The curves x^x (dipping to a minimum) and x^{−x} on [0,1]; their areas are the two n^n series.
4 TWO DIMENSIONS · INTERACTIVE
Add series terms; the n^n partial sums converge to the integrals ∫₀¹ x^x dx and ∫₀¹ x^{−x} dx.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two integral values, equal to their n^n series.
AVAN’s addition (the inverse-companion): don’t integrate the self-power — expand it. The inverse of ‘∫01 x±x dx’ is ‘the series ∑ (±1)n-1/nn’, obtained by expanding e±x ln x and integrating term by term. Magenta are the self-power curves; green are the nn series they equal. A self-power integral read as a clean series.
LIT Genuine sophomore's dream (Johann Bernoulli, 1697). Verified live: the numerical integrals ∫₀¹ x^x dx and ∫₀¹ x^{−x} dx match the series Σ(−1)^{n−1}/nⁿ and Σ1/nⁿ respectively to ~1e-6 (window.__sophomore.okA, .okB, .Ip, .In).
FIG No framing; the numerical integrals and the self-power series are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of integrating the self-power, expand it: the inverse of '∫₀¹ x^{±x} dx' is 'the series Σ(±1)^{n−1}/nⁿ', obtained by expanding e^{±x ln x} and integrating term by term. Magenta are the self-power curves; green are the nⁿ series they equal. A self-power integral read as a clean series.
FIG No framing; the numerical integrals and the self-power series are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of integrating the self-power, expand it: the inverse of '∫₀¹ x^{±x} dx' is 'the series Σ(±1)^{n−1}/nⁿ', obtained by expanding e^{±x ln x} and integrating term by term. Magenta are the self-power curves; green are the nⁿ series they equal. A self-power integral read as a clean series.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN