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

The two sequences differ by a null term

Consider the sequences $(a_{n})_{n} \geq_{1}$ and $(b_{n})_{n} \geq_{1}$ , defined by $a_{n} = - 2 n + \sum (k = 1.. n) 1/ k$ and $b_{n} = - 2 n + 1 + \sum (k = 1.. n) 1/ k$ . Which of the following statements is true?

A. $(a_{n})_{n} \geq_{1}$ converges but $(b_{n})_{n} \geq_{1}$ does not converge.
B. $(b_{n})_{n} \geq_{1}$ converges but $(a_{n})_{n} \geq_{1}$ does not converge.
C.Both $(a_{n})_{n} \geq_{1}$ and $(b_{n})_{n} \geq_{1}$ converge.✓
D.Neither $(a_{n})_{n} \geq_{1}$ nor $(b_{n})_{n} \geq_{1}$ 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.

See pricing

50 are analysed free — try those first.

Related counterexample: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

The two sequences differ by a null term

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