Consider the series Σ{n=1}^∞ aₙ, where aₙ = (−1)^{n+1}(√(n+1) − √n). Which of the following statements is true?

C. √(n+1) − √n = 1/(√(n+1) + √n) decreases to 0, so the alternating series converges (Leibniz). The absolute series behaves like Σ 1/(2√n), which diverges. Hence conditionally convergent — option 3 is the most precise true statement (official key: 3).

Part BCSIR NET June 2023conditional-vs-absolute

Conditional vs absolute

Consider the series $\sum_{n=1}^\infty$ $a_{n}$ , where $a_{n} =$ $(-1)^{n+1}(\sqrt{n+1} - \sqrt{n})$ . Which of the following statements is true?

A.The series is divergent.
B.The series is convergent.
C.The series is conditionally convergent.✓ Correct
D.The series is absolutely convergent.

The idea, in plain words

The difference of two nearby square roots shrinks like one over a square root, which is slow — too slow to add up on its own. But the terms alternate in sign, and alternating terms that shrink to zero always settle down, so it converges without converging absolutely.

Solution

$\sqrt{n+1} - \sqrt{n} = 1/(\sqrt{n+1} + \sqrt{n})$ decreases to 0

so the alternating series converges (Leibniz).

The absolute series behaves like $\sum 1/(2\sqrt{n})$

which diverges.

Hence conditionally convergent — option 3 is the most precise true statement (official key: 3).

Why each option is right or wrong

Checked against 3 symbolic computations

A.Wrong
False.

Rationalising, $\sqrt{n+1} - \sqrt{n} = 1/(\sqrt{n+1} + \sqrt{n})$ — SymPy confirms the identity exactly.

That expression decreases monotonically to 0 (verified)

so the alternating series test applies and the series converges.
B.WrongBoundary and endpoint
Not the intended answer, and it is worth being clear why: this statement is literally true — the series does converge.

But 'conditionally convergent' in (C) is strictly more informative, and the examiner marks only the most precise true statement.

When two options are nested like this, always choose the sharper one.

If you found (B) attractive, your mathematics was right and your reading of the instruction was not.
C.Correct
True, and it is the sharpest correct description.

Convergence: the terms $1/(\sqrt{n+1} + \sqrt{n})$ decrease to 0

so Leibniz applies.

Non-absolute convergence: | $a_{n}$ | $= 1/(\sqrt{n+1} + \sqrt{n})$ behaves like $1/(2\sqrt{n})$ , and SymPy's sampling shows that decaying only as slowly as 0.005 by $t = 10^{4}$ — a p-series with p = ½

which diverges.

Convergent but not absolutely convergent is exactly 'conditionally convergent'.
D.WrongExecution slip
False.

Absolute convergence would require $\sum1/(\sqrt{n+1} + \sqrt{n})$ to converge

but that series is comparable to $\sum1/(2\sqrt{n}), a$ divergent p-series with p = ½ ≤ 1.

The rationalisation is what exposes the true size of the terms — without it, $\sqrt{n+1} - \sqrt{n}$ looks deceptively small.

2 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: Σaₙ convergent ⇒ Σaₙ² convergent

Conditional vs absolute

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