LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-16

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The binomial double series for re-expanding a complex power series is absolutely convergent and may be regrouped

Statement

Let $\sum c_{n} (z - a)^{n}$ have radius $R$ . If $∣ b - a ∣ + ∣ h ∣ < R$ , then $\sum_{n \geq 0} \sum_{k = 0}^{n} (\binom{n}{k}) ∣ c_{n} ∣ ∣ b - a ∣^{n - k} ∣ h ∣^{k} < \infty,$ and the complex binomial double series may be regrouped by powers of $h$ .

Facts & Assumptions

Given: A complex power series and points $b, h$ satisfying $∣ b - a ∣ + ∣ h ∣ < R$ .

[L1]

The corresponding nonnegative real binomial double series converges and licenses regrouping (The binomial double series used to re-expand a power series at an interior point is absolutely convergent and may be regrouped).

[L2]

For complex $z, w$ and $n \in N$ , $(z + w)^{n} = \sum_{k \leq n} (\binom{n}{k}) z^{k} w^{n - k}$ (The binomial theorem over the complex field).

[L3]

Every absolutely convergent complex series converges, and every rearrangement has the same sum (Every absolutely convergent complex series converges, and rearrangements preserve its sum).

Proof

technique · direct

1.1L1

Apply [L1] to the real coefficient sequence $∣ c_{n} ∣$ and the nonnegative numbers $∣ b - a ∣, ∣ h ∣$ ; this gives the displayed finite total majorant.

2.1step 1.1L2

By [L2], $c_{n} ((b - a) + h)^{n}$ is the finite sum over $k \leq n$ of the corresponding complex terms, each bounded by the majorant term in step 1.1.

3.1step 1.1step 2.1L3∎

Absolute convergence now permits regrouping by $k$ under [L3]. The cases $h = 0$ and $b = a$ merely make some terms vanish and are included.

Depends on

Used by

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

Dependency tree · two levels

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