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Part BCSIR NET December 2024the-two-sequences-differ-by-a-null-term

The two sequences differ by a null term

Consider the sequences (an)n≥1(a_{n})_{n}\ge_{1} and (bn)n≥1(b_{n})_{n}\ge_{1}, defined by an=−2n+∑(k=1..n)1/ka_{n} = -2\sqrt{n} + \sum(k=1..n) 1/\sqrt{k} and bn=−2n+1+∑(k=1..n)1/kb_{n} = -2\sqrt{n+1} + \sum(k=1..n) 1/\sqrt{k}. Which of the following statements is true?

  1. A.(an)n≥1(a_{n})_{n}\ge_{1} converges but (bn)n≥1(b_{n})_{n}\ge_{1} does not converge.
  2. B.(bn)n≥1(b_{n})_{n}\ge_{1} converges but (an)n≥1(a_{n})_{n}\ge_{1} does not converge.
  3. C.Both (an)n≥1(a_{n})_{n}\ge_{1} and (bn)n≥1(b_{n})_{n}\ge_{1} converge.✓ Correct
  4. D.Neither (an)n≥1(a_{n})_{n}\ge_{1} nor (bn)n≥1(b_{n})_{n}\ge_{1} converges.

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 limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

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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