LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

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The binomial double series used to re-expand a power series at an interior point is absolutely convergent and may be regrouped

Statement

Let $\sum_{n \geq 0} a_{n} (x - c)^{n}$ have radius $R$ , let $d$ satisfy $∣ d - c ∣ < R$ , and let $h$ satisfy $∣ d - c ∣ + ∣ h ∣ < R$ . Then

$\sum_{n = 0}^{\infty} \sum_{k = 0}^{n} ι ⁣ (\binom{n}{k}) ∣ a_{n} ∣ ∣ d - c ∣^{n - k} ∣ h ∣^{k} < \infty .$

Consequently the binomial double series is absolutely convergent and may be regrouped by powers of $h$ :

$\sum_{n = 0}^{\infty} a_{n} (d + h - c)^{n} = \sum_{k = 0}^{\infty} (\sum_{n = k}^{\infty} ι ⁣ (\binom{n}{k}) a_{n} (d - c)^{n - k}) h^{k} .$

Facts & Assumptions

Given: The power series and points $c, d, d + h$ from the statement.

[L1]

A power series converges absolutely at every distance smaller than its radius (A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint).

[L2]

The binomial theorem gives $\sum_{k = 0}^{n} ι ⁣ (\binom{n}{k}) u^{n - k} v^{k} = (u + v)^{n}$ for all real $u, v$ (The binomial theorem in $R$ : $(x + y)^{n} = \sum_{k < n + 1} ι ⁣ (\binom{n}{k}) x^{k} y^{n - k}$ ).

[L3]

An absolutely convergent double series may be summed and regrouped in either order (Fubini for double series: if $\sum_{i} \sum_{j} ∣ a_{i j} ∣$ converges then both iterated sums and the sum along every bijection $N \to N \times N$ converge to one and the same value, Absolutely convergent and conditionally convergent series, and the general starting index).

Proof

technique · direct

1.1

Put $ρ : = ∣ d - c ∣ + ∣ h ∣ < R$ . By [L2], the sum of the absolute values in row $n$ is $∣ a_{n} ∣ ρ^{n}$ .

givenL2algebra

2.1

The series $\sum_{n} ∣ a_{n} ∣ ρ^{n}$ converges by [L1], so the triangular double series is absolutely convergent.

step 1.1L1

3.1

Apply the binomial theorem before summing and [L3] to regroup the absolutely convergent double series by $k$ . This yields the displayed identity.

step 2.1L2L3∎

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Sources

Power series, Encyclopedia of Mathematics (standard reference, not scraped)
Northwestern Math 320-2 lecture notes (standard reference, not scraped)