PropositionStatement: AI-adaptedProof: 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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Sums and scalar multiples of convergent complex power series are represented coefficientwise on the common disc

Statement

If $f (z) = \sum a_{n} (z - c)^{n}$ and $g (z) = \sum b_{n} (z - c)^{n}$ , then for complex scalars $α, β$ , $α f (z) + β g (z) = \sum_{n \geq 0} (α a_{n} + β b_{n}) (z - c)^{n}$ throughout the common open disc of convergence, with local uniform convergence there.

Facts & Assumptions

Given: Two complex power series about the same centre and scalars $α, β$ .

[L1]

Each series converges uniformly on every smaller closed subdisc (A complex power series converges absolutely and uniformly on every closed subdisc strictly inside its disc of convergence).

[L2]

Complex modulus satisfies $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ and $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1

For every finite $N$ , exact distributivity gives $α \sum_{n < N} a_{n} (z - c)^{n} + β \sum_{n < N} b_{n} (z - c)^{n} = \sum_{n < N} (α a_{n} + β b_{n}) (z - c)^{n}$ .

algebra

1.2

On a closed subdisc inside both radii, [L1] makes both sequences of partial sums uniformly convergent. If their limits are $A, B$ , then [L2] bounds the error after taking the linear combination by $∣ α ∣ ∣ A_{N} - A ∣ + ∣ β ∣ ∣ B_{N} - B ∣$ , which tends uniformly to $0$ .

L1L2algebra

2.1

Passing to the limit in step 1.1 proves the formula and its local uniform convergence. Zero scalars and unequal radii are included by taking the common disc.

step 1.1step 1.2∎

Depends on

Used by

Complex analytic functions are closed under finite linear combinations, products, quotients with nonzero denominator, and composition Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 11 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

MIT 18.100C lecture notes on power series (standard reference, not scraped)