The alternating harmonic series ∑(−1)ⁿ⁺¹/n converges. A student rearranges its terms to group all positive terms first, then all negative terms. What is the most accurate statement about the sum after rearrangement?
AThe sum is unchanged, because addition is commutative
BThe sum may differ — the series converges conditionally, so rearrangements can change or destroy convergence
CThe sum doubles, because the positive terms are now grouped
DThe rearrangement diverges, because the alternating sign was removed
The alternating harmonic series converges conditionally (it converges, but ∑1/n diverges). The Riemann Rearrangement Theorem states that a conditionally convergent series can be rearranged to converge to any real number — or to diverge. Commutativity of addition applies to finite sums, not to conditionally convergent infinite series. If the series converged absolutely, the sum would be rearrangement-invariant.
Question 2 Multiple Choice
To classify a series as absolutely convergent, conditionally convergent, or divergent, what is the correct two-step procedure?
AApply the alternating series test first; if it passes, the series converges absolutely
BTest ∑|aₙ| for convergence first; if it converges, the series is absolutely convergent. If ∑|aₙ| diverges but ∑aₙ converges, it is conditionally convergent
CTest ∑aₙ for convergence first; if it converges, test ∑|aₙ|. If ∑|aₙ| also converges, it is conditionally convergent
DUse the ratio test on ∑aₙ; if L < 1 the series is absolutely convergent, if L = 1 it is conditionally convergent
The correct order: first test ∑|aₙ|. Absolute convergence (∑|aₙ| converges) is the stronger condition and implies ordinary convergence. If ∑|aₙ| diverges but ∑aₙ converges (typically confirmed by the alternating series test), the convergence is conditional. Option C reverses the logic — 'conditionally convergent' means ∑aₙ converges but ∑|aₙ| diverges, not the other way around.
Question 3 True / False
If a series converges absolutely, then it also converges in the ordinary sense.
TTrue
FFalse
Answer: True
This is the key implication: absolute convergence ⟹ convergence. The proof shows that if ∑|aₙ| converges, the partial sums of ∑aₙ form a Cauchy sequence and must converge. The implication goes only one way — a series can converge without converging absolutely (conditional convergence), so convergence does NOT imply absolute convergence.
Question 4 True / False
If a series ∑aₙ converges, then the series ∑|aₙ| also converges.
TTrue
FFalse
Answer: False
This is the most common confusion about the two types of convergence. The alternating harmonic series is the standard counterexample: ∑(−1)ⁿ⁺¹/n converges (by the alternating series test), but ∑1/n is the harmonic series, which diverges. So the series converges without converging absolutely. The implication 'convergence ⟹ absolute convergence' is false; only the reverse holds.
Question 5 Short Answer
Explain why the Riemann Rearrangement Theorem applies to conditionally convergent series but cannot apply to absolutely convergent ones.
Think about your answer, then reveal below.
Model answer: For a conditionally convergent series, the positive terms alone diverge to +∞ and the negative terms alone diverge to −∞. By selectively interleaving positive and negative terms, you can overshoot or undershoot any target sum by any amount and then correct course, allowing you to converge to any desired value. For an absolutely convergent series, the total 'mass' of positive terms and the total 'mass' of negative terms are each finite — there is no infinite reservoir to draw on. No matter how you rearrange, you're redistributing a fixed total, so the sum is invariant.
The key is that conditional convergence arises from cancellation between infinite positive and negative reservoirs. Absolute convergence means the series converges on magnitude alone — sign-dependent cancellation is not the mechanism. This structural difference is what makes rearrangement dangerous for conditionally convergent series and harmless for absolutely convergent ones.