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Part BCSIR NET December 2025alpha-less-than-one-diverges-regardless-of-beta-only-alpha-equals-one-hands-the-question-to-beta

Alpha less than one diverges regardless of beta only alpha equals one hands the question to beta

Let α,β∈(0,∞)\alpha, \beta \in (0, \infty). Consider the infinite series ∑n≥4\sum_{n\ge4} 1/[n(logen)α(loge(logen))β]1/[n(log_{e} n)^\alpha (log_{e}(log_{e} n))^\beta]. Which of the following statements is true?

  1. A.The series converges for all α∈(0,1)\alpha \in (0,1) and for all β∈(1,∞)\beta \in (1,\infty).
  2. B.The series converges for α=1\alpha = 1 and β=1\beta = 1.
  3. C.The series converges for α=1\alpha = 1 and for all β∈(1,∞)\beta \in (1,\infty).✓
  4. D.The series converges for all α∈(0,∞)\alpha \in (0,\infty) and for all β∈(0,∞)\beta \in (0,\infty).

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.

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50 are analysed free — try those first.

Related counterexample: Σaₙ convergent ⇒ Σaₙ² convergent

More on this topic

The chapter behind this: Series tests — decision tree and the trap list — free to read

From The Real Line › Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

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