NETMaths
Part CCSIR NET December 2023p-integral-thresholds

P integral thresholds

For a real number , consider the improper integrals dx and dx. Which of the following statements are true?

  1. A.There exists such that converges, but does not converge.
  2. B.There exists such that converges, but does not converge.
  3. C.There exists such that both converge.
  4. D.There exists such that neither nor converges.

Solution

converges iff converges iff . So they never converge together, and at neither does.

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: Riemann integrable ⇒ continuous almost everywhere fails for… (i.e. 'bounded ⇒ integrable')

More on this topic

From IntegrationRiemann integration and criteria

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