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Part CCSIR NET June 2024unbounded-set-forces-infinite-radius

Unbounded set forces infinite radius

Which of the following conditions ensure that the power series ∑(n≥0)anzn\sum_(n\ge0) a_{n}z^{n} defines an entire function?

  1. A.The power series converges for every z∈Cz \in \mathbb{C}✓ Correct
  2. B.The power series converges for every z∈Rz \in \mathbb{R}✓ Correct
  3. C.The power series converges for every z ∈ {2n:n∈N2^{n} : n \in \mathbb{N}}✓ Correct
  4. D.The power series converges for every z ∈ {1/5n:n∈N1/5^{n} : n \in \mathbb{N}}

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, and why this one tests boundary and endpoint.

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50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Zeros of a non-constant holomorphic function cannot accumulate

More on this topic

The chapter behind this: Analyticity, the identity theorem and zeros — free to read

From Analytic Functions › Power series and analyticity

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