TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-07-31

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A power-series sum may be re-expanded about every interior point, with coefficients given by its derivatives there

Statement

Suppose $f (x) = \sum_{n \geq 0} a_{n} (x - c)^{n}$ has radius $R$ , and let $d$ satisfy $∣ d - c ∣ < R$ . Then for every real $x$ with

$∣ d - c ∣ + ∣ x - d ∣ < R$

one has

$f (x) = \sum_{k = 0}^{\infty} \frac{f^{(k)} (d)}{ι (k!)} (x - d)^{k} .$

Thus the sum may be re-expanded about every interior point.

Facts & Assumptions

Given: The series for $f$ and the interior point $d$ .

[L1]

The binomial double series is absolutely convergent when $∣ d - c ∣ + ∣ x - d ∣ < R$ and may be regrouped by powers of $x - d$ (The binomial double series used to re-expand a power series at an interior point is absolutely convergent and may be regrouped).

[L2]

Repeated termwise differentiation gives $f^{(k)} (d) = \sum_{n \geq k} ι (n^{\underline{k}}) a_{n} (d - c)^{n - k}$ (A power-series sum is infinitely differentiable inside its radius and satisfies $a_{n} = f^{(n)} (c) / ι (n!)$ at its centre).

[L3]

$(\binom{n}{k}) k! = n^{\underline{k}}$ for $k \leq n$ ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ).

Proof

technique · direct

1.1

Fix $x$ satisfying the stated inequality and set $h : = x - d$ . By [L1], $f (x) = \sum_{k \geq 0} b_{k} h^{k}$ , where $b_{k} : = \sum_{n \geq k} ι ⁣ (\binom{n}{k}) a_{n} (d - c)^{n - k}$ .

givenL1

2.1

By [L2] and [L3], $ι (k!) b_{k} = f^{(k)} (d)$ for every $k$ .

step 1.1L2L3algebra

3.1

Substituting the coefficient identity from step 2.1 into the series in step 1.1 proves the formula.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

MIT 18.100C, Lecture 11: Power Series (standard reference, not scraped)
Power series, Encyclopedia of Mathematics (standard reference, not scraped)
Northwestern Math 320-2 lecture notes (standard reference, not scraped)