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Part CCSIR NET December 2024rationalise-and-the-sequence-becomes-monotone

Rationalise and the sequence becomes monotone

Let A = {1/(n+n−n)1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) | n∈Z,n>0n \in \mathbb{Z}, n > 0}. Which of the following statements are true?

  1. A.sup A is finite.✓ Correct
  2. B.lim⁡sup⁡(n→∞)1/(n+n−n)<sup⁡A\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) < \sup A✓ Correct
  3. C.lim⁡inf⁡(n→∞)1/(n+n−n)=2\lim \inf(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 2✓ Correct
  4. D.lim⁡sup⁡(n→∞)1/(n+n−n)=3\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 3

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: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

More on this topic

The chapter behind this: Sequences in ℝ — the implication map — free to read

From The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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