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Part BCSIR NET June 2025cauchy-hadamard-tests-the-coefficients-so-divide-by-n-factorial-first

Cauchy hadamard tests the coefficients so divide by n factorial first

Which of the following statements is true?

  1. A.There exists an entire function f such that f⁽n^{n}⁾(0)=n!/nn(0) = n!/n^{n} for all positive integers n.✓ Correct
  2. B.There exists an entire function f such that f⁽n^{n}⁾(0)=n!nn(0) = n!n^{n} for all positive integers n.
  3. C.There exists an entire function f such that f⁽n^{n}⁾(0) = (n − 1)! for all positive integers n.
  4. D.There exists an entire function f such that f⁽n^{n}⁾(0) = n!n for all positive integers 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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

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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