Part CCSIR NET December 2023hurwitz-zeros-escape

Hurwitz zeros escape

For every n ≥ 1, consider the entire function $p_{n}(z) = \sum_{k=0}^{n}$ zᵏ/k!. Which of the following statements are true?

A.The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $\mathbb{C}$ .✓ Correct
B.For all $n \ge 1, p_{n}$ has a zero in the set { $z \in \mathbb{C}$ : |z| ≤ 2023}.
C.There exists a sequence $(z_{n})$ of complex numbers such that lim | $z_{n}$ | $= \infty$ and $p_{n}(z_{n}) = 0$ for all n ≥ 1.✓ Correct
D.Let $S_{n}$ be the set of all zeros of $p_{n}$ . If $a_{n} =$ $min_{z\in{}S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$ .✓ Correct

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 standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Hurwitz zeros escape

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