CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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The series sum z to the n over n squared is continuous on the closed disc but singular at 1

Statement refuted

Refuted claim. If a power-series sum extends continuously to the closed disc of convergence, then every boundary point is regular.

The witness is

$f (z) = \sum_{n \geq 1} \frac{z^{n}}{n^{2}} .$

This sum is continuous on $\overline{D}$ but the boundary point $1$ is singular.

Facts & Assumptions

Given: The series $f (z) = \sum_{n \geq 1} z^{n} / n^{2}$ .

[L1]

The $p$ -series $\sum_{n \geq 1} 1 / n^{2}$ converges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ).

[L2]

A convergent numerical majorant makes a complex function series converge uniformly on the domain of the bound (Weierstrass M-test for complex-valued function series).

[L3]

The coefficients $1 / n^{2}$ give radius $1$ , and Pringsheim makes the positive boundary point singular because those coefficients are nonnegative (Cauchy-Hadamard for complex power series, including zero and infinite radius, Pringsheim's theorem for power series with nonnegative coefficients).

Counterexample

technique · direct

1.1

On the closed unit disc one has $∣ z^{n} / n^{2} ∣ \leq 1 / n^{2}$ . By [L1] and [L2], the series for $f$ converges uniformly on $\overline{D}$ , so its sum is continuous there.

L1L2

2.1

Fact [L3] gives radius $1$ and shows that the boundary point $1$ is singular. Therefore continuity on the closed disc does not force regularity at all boundary points, and the displayed claim is false.

L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

Philippe Flajolet, Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis, Theorem 4 (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Ch. 4 (standard reference, not scraped)