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

Unbounded set forces infinite radius

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

A.The power series converges for every $z \in C$ ✓
B.The power series converges for every $z \in R$ ✓
C.The power series converges for every z ∈ { $2^{n} : n \in N$ }✓
D.The power series converges for every z ∈ { $1/ 5^{n} : n \in N$ }

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.

50 are analysed free — try those first.

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

More on this topic

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only multiples of four surviveDecember 2024

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

From Analytic Functions › Power series and analyticity

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