ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The geometric series represents $1 / (1 - x)$ for $∣ x ∣ < 1$ and re-expands explicitly about every $c$ with $∣ c ∣ < 1$

Statement

For $∣ x ∣ < 1$ ,

$\frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n} .$

More generally, if $∣ c ∣ < 1$ , then

$\frac{1}{1 - x} = \sum_{n = 0}^{\infty} \frac{(x - c)^{n}}{(1 - c)^{n + 1}} (∣ x - c ∣ < 1 - c) .$

Facts & Assumptions

Given: A real $c$ with $∣ c ∣ < 1$ .

[L1]

The geometric series sums to $1 / (1 - t)$ for $∣ t ∣ < 1$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[L2]

Power-series sums re-expand about interior points, and a nonzero local denominator has a reciprocal series (A power-series sum may be re-expanded about every interior point, with coefficients given by its derivatives there, A convergent real power series with nonzero constant term has a convergent reciprocal power series on a smaller neighbourhood).

Verification

technique · direct

1.1

Apply [L1] with $t = x$ to get the first formula.

2.1

The general results in [L2] show qualitatively that the sum re-expands about $c$ and that the nonzero denominator there has a local reciprocal series. To identify that series and its full convergence interval directly, use $1 - x = (1 - c) (1 - (x - c) / (1 - c))$ and $1 - c > 0$ , and apply [L1] with $t = (x - c) / (1 - c)$ . This gives the second formula precisely when $∣ x - c ∣ < 1 - c$ .

givenL1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Power series, Encyclopedia of Mathematics (standard reference, not scraped)
Northwestern Math 320-2 lecture notes (standard reference, not scraped)

The geometric series represents 1/(1−x)1/(1−x) for ∣x∣<1∣x∣<1 and re-expands explicitly about every cc with ∣c∣<1∣c∣<1