LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1

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

algebra

2.1

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.

step 1.1L1

3.1

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

step 2.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 89 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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