LemmaStatement: AI-adaptedProof: AI-adaptedprecheck 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.
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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 $F (y) = \sum_{m \geq 0} b_{m} (y - e)^{m}$ have positive radius $S$ , and let $G (x) - e = \sum_{n \geq 1} c_{n} (x - d)^{n}$ converge near $d$ . If some $r > 0$ satisfies

$B_{r} : = \sum_{n \geq 1} ∣ c_{n} ∣ r^{n} < S,$

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 $A B$ ).

[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 $\sum_{i} \sum_{j} ∣ a_{i j} ∣$ converges then both iterated sums and the sum along every bijection $N \to N \times N$ converge to one and the same value).

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 $∣ x - d ∣ \leq r$ , 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 $∣ x - d ∣ \leq r$ , 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 · two levels

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Sources

Power series, Encyclopedia of Mathematics (standard reference, not scraped)
E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)