TheoremStatement: 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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A complex power-series sum re-expands about every interior point, at least to the distance from that point to the original boundary

Statement

If $f (z) = \sum_{n \geq 0} c_{n} (z - a)^{n}$ has radius $R$ and $∣ b - a ∣ < R$ , then $f (b + h) = \sum_{k \geq 0} \frac{f^{(k)} (b)}{k!} h^{k} whenever ∣ h ∣ < R - ∣ b - a ∣ .$ The displayed bound is a guaranteed radius, not necessarily the exact radius of the new series.

Facts & Assumptions

Given: A complex power-series sum $f$ and an interior point $b$ .

[L1]

Under $∣ b - a ∣ + ∣ h ∣ < R$ , the binomial double series is absolutely convergent and may be regrouped (The binomial double series for re-expanding a complex power series is absolutely convergent and may be regrouped).

[L2]

The coefficient of a representation about $b$ is $f^{(k)} (b) / k!$ (The coefficients of a complex power series are its derivatives at the centre divided by the corresponding factorials).

[L3]

Every derivative of a power-series sum is obtained termwise (A complex power-series sum has complex derivatives of every order, obtained by repeated termwise differentiation).

Proof

technique · direct

1.1

Put $z = b + h$ . The binomial expansion and [L1] give $f (b + h) = \sum_{k \geq 0} d_{k} h^{k}$ , where $d_{k} = \sum_{n \geq k} (\binom{n}{k}) c_{n} (b - a)^{n - k}$ .

L1algebra

2.1

By [L3], evaluating the $k$ th derivative at $b$ gives $f^{(k)} (b) = k! d_{k}$ .

step 1.1L3

3.1

By [L2], $d_{k} = f^{(k)} (b) / k!$ , proving the stated expansion for $∣ h ∣ < R - ∣ b - a ∣$ . The bound remains valid when $R = + \infty$ .

step 1.1step 2.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 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

Power-series re-expansion notes, Colby College (standard reference, not scraped)