Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31
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The sum of a real power series is real analytic throughout the open interval determined by its radius

Statement

If a real power series centred at cc has radius R>0R>0, then its sum is real analytic on (c−R,c+R)(c-R,c+R), interpreted as all of R\mathbb R when R=+∞R=+\infty.

Facts & Assumptions

Given: A power-series sum ff on its open radius interval.

[L1]

At every point dd strictly inside the radius, ff has a convergent re-expansion in powers of x−dx-d on a positive neighbourhood (A power-series sum may be re-expanded about every interior point, with coefficients given by its derivatives there).

[L2]

Real analyticity means precisely the existence of such a local power-series representation at every point (A real-analytic function on an open subset of R\mathbb{R} is locally represented by a convergent real power series).

Proof

technique · direct
1.1

Fix dd in the open radius interval. Then R−∣d−c∣>0R-|d-c|>0, and [L1] represents ff by a power series about dd whenever ∣x−d∣<R−∣d−c∣|x-d|<R-|d-c|.

givenL1
2.1

The neighbourhood in step 1.1 lies inside the open radius interval, so [L2] applies at every dd and proves that ff is real analytic there.

step 1.1L2∎

Depends on

Used by

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Sources