LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A composition of convergent real power series has a convergent power-series expansion wherever the inner series maps a neighbourhood into the outer disk of convergence

Statement

Let $F(y)=\sum_{m\ge0}b_m(y-e)^m$ have positive radius $S$ , and let $G(x)-e=\sum_{n\ge1}c_n(x-d)^n$ converge near $d$ . If some $r>0$ satisfies

$B_r:=\sum_{n\ge1}|c_n|r^n<S,$

then $F\circ G$ is represented for $|x-d|<r$ by a convergent power series about $d$ , obtained by expanding and regrouping $\sum_m b_m(G(x)-e)^m$ .

Facts & Assumptions

Given: The outer and inner series and $r$ from the statement.

[L1]

The Cauchy product of two absolutely convergent series converges absolutely, and its absolute sum is at most the product of the two absolute sums (If $\sum a_k$ and $\sum b_k$ both converge absolutely then their Cauchy product converges absolutely, with sum $AB$ ).

[L2]

The outer series converges absolutely at every distance below $S$ (A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint).

[L3]

An absolutely convergent multiple series may be regrouped (Fubini for double series: if $\sum_i \sum_j |a_{ij}|$ converges then both iterated sums and the sum along every bijection $\mathbb{N} \to \mathbb{N} \times \mathbb{N}$ converge to one and the same value).

Proof

technique · constructive

1.1

For each $m$ , repeatedly use [L1] to expand $(G(x)-e)^m$ in powers of $x-d$ ; take the zeroth power to be $1$ . For $|x-d|\le r$ , the inner numerical series is absolutely convergent with absolute sum at most $B_r$ , so the expanded $m$ th power has absolute term sum at most $B_r^m$ .

givenconstructL1

2.1

Consequently, for $|x-d|\le r$ , the sum of absolute values of all expanded terms with outer degree $m$ is at most $|b_m|B_r^m$ . The series of these bounds converges because $B_r<S$ and [L2] applies.

step 1.1L2algebra

3.1

By [L3], regroup the absolutely convergent expansion by total powers of $x-d$ . The resulting power series converges on $|x-d|<r$ and sums to $\sum_m b_m(G(x)-e)^m=F(G(x))$ .

step 1.1step 2.1L3discharge-construct∎

Depends on

Used by

Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 105 results over 22 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, Encyclopedia of Mathematics (standard reference, not scraped)
E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)