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Part BCSIR NET December 2025vanishing-at-infinity-on-a-once-punctured-plane-forces-genuine-o-of-1-over-z-decay-not-just-a-limit

Vanishing at infinity on a once punctured plane forces genuine o of 1 over z decay not just a limit

Let h be a holomorphic function on C\mathbb{C}{0} such that lim∣z∣→∞lim_{|z|\to\infty} h(z) = 0. For every n ≥ 1, consider fn(z)=∑k=1nf_{n}(z) = \sum_{k=1}^n h(zᵏ). Let 𝔻 denote the open unit disc {z∈Cz \in \mathbb{C} : |z| < 1}. Which of the following statements is necessarily true?

  1. A.For all z with |z| ≥ 1, the sequence {fn(z)f_{n}(z)}n≥1_{n}\ge1 converges.
  2. B.For all z ∈ 𝔻{0}, the sequence {fn(z)f_{n}(z)}n≥1_{n}\ge1 converges.
  3. C.The sequence {fnf_{n}}n≥1_{n}\ge1 converges pointwise to a holomorphic function on 𝔻[0,1).
  4. D.The sequence {fnf_{n}}n≥1_{n}\ge1 converges pointwise to a holomorphic function on {z∈Cz \in \mathbb{C} : |z| > 1}.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: |f| bounded near an isolated singularity ⇒ pole

More on this topic

The chapter behind this: Singularities — three types, three tests — free to read

From Singularities and Residues › Laurent series, classification of singularities, Casorati–Weierstrass

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