LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

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.

A complex power series, its formal derivative, and its zero-constant-term formal antiderivative have the same radius

Statement

The complex power series $\sum_{n \geq 0} c_{n} (z - a)^{n}$ , its formal derivative $\sum_{n \geq 0} (n + 1) c_{n + 1} (z - a)^{n}$ , and its zero-constant-term formal antiderivative $\sum_{n \geq 0} c_{n} (z - a)^{n + 1} / (n + 1)$ have the same radius of convergence.

Facts & Assumptions

Given: A complex power series with coefficients $(c_{n})$ .

[L1]

A complex power series converges absolutely exactly when the corresponding real modulus-coefficient series converges (Complex series, absolute convergence, complex power series, and radius of convergence).

[L2]

A real power series, its formal derivative, and its zero-constant-term formal antiderivative have the same radius (A power series, its formal derivative, and its zero-constant-term formal antiderivative have the same radius of convergence).

Proof

technique · direct

1.1algebra

The modulus coefficient of the formal derivative is $(n + 1) ∣ c_{n + 1} ∣$ , and that of the formal antiderivative is $∣ c_{n} ∣ / (n + 1)$ .

2.1step 1.1L1

By [L1], the three complex radii are precisely the radii of the three real power series with the modulus coefficients described in step 1.1.

3.1step 2.1L2∎

Apply [L2] to those real series. The conclusion includes radii $0$ and $+ \infty$ and uses ordinary embedded-number notation only.

Depends on

Used by

Dependency tree · two levels

18 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 1, §2.3 (standard reference, not scraped)