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

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A power series converges absolutely and uniformly on every closed interval strictly inside its interval of convergence

Statement

Let $\sum a_{n} (x - c)^{n}$ have radius $R$ , and let $[u, v]$ be a nonempty closed interval for which

$r : = \max {∣ u - c ∣, ∣ v - c ∣} < R .$

Then the function series converges absolutely at every point of $[u, v]$ and converges uniformly there.

Facts & Assumptions

Given: A power series of radius $R$ and a closed interval $[u, v]$ satisfying the strict interior condition above (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A series of real-valued functions and its pointwise and uniform convergence through its partial sums).

[L1]

The power series converges absolutely at every point whose distance from $c$ is less than $R$ (A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint).

[L2]

If $∣ f_{n} (x) ∣ \leq M_{n}$ for all $n, x$ and $\sum M_{n}$ converges, the Weierstrass M-test gives absolute pointwise and uniform convergence of $\sum f_{n}$ (The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series).

Proof

technique · direct

1.1

Choose a real $ρ$ with $r < ρ < R$ , or merely $ρ > r$ when $R = + \infty$ . Then the scalar series $\sum ∣ a_{n} ∣ ρ^{n}$ converges by [L1], applied at $x = c + ρ$ .

givenL1choose

1.2

For every $x \in [u, v]$ , order-convexity gives $∣ x - c ∣ \leq \max {∣ u - c ∣, ∣ v - c ∣} = r < ρ$ , and hence $∣ a_{n} (x - c)^{n} ∣ \leq ∣ a_{n} ∣ ρ^{n}$ for every $n$ .

givenalgebra

2.1

Apply [L2] to $f_{n} (x) = a_{n} (x - c)^{n}$ and $M_{n} = ∣ a_{n} ∣ ρ^{n}$ . The series is absolutely convergent at each $x \in [u, v]$ and uniformly convergent on the whole interval.

step 1.1step 1.2L2∎

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MIT 18.100C, Lecture 11: Power Series (standard reference, not scraped)
Power series, Encyclopedia of Mathematics (standard reference, not scraped)