The alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + ⋯ converges to ln 2. If the terms are rearranged so that two positive terms always precede one negative term (e.g., 1 + 1/3 − 1/2 + 1/5 + 1/7 − 1/4 + ⋯), what can we conclude about the rearranged series?
AIt still converges to ln 2, because the same terms are present
BIt diverges, because the terms are out of order
CIt converges to a different value, approximately (3/2) ln 2
DIt cannot converge because rearranging an infinite series always destroys convergence
This is an instance of the Riemann rearrangement theorem: a conditionally convergent series can be rearranged to converge to any real value. The alternating harmonic series is conditionally convergent (∑1/n diverges, so it is not absolutely convergent), so rearrangements can produce different sums. The two-positive-one-negative rearrangement converges to (3/2) ln 2 ≈ 1.04 rather than ln 2 ≈ 0.69. Option A is the most tempting error — the naive intuition that 'same terms = same sum' fails for infinite series when convergence is only conditional.
Question 2 Multiple Choice
Which of the following series can safely be rearranged into any order without changing its sum?
A∑(−1)ⁿ/n (the alternating harmonic series)
B∑(−1)ⁿ/n² (an alternating series whose absolute value also converges)
C∑(−1)ⁿ (the alternating series of ±1, which diverges)
D∑1/n (the harmonic series, which diverges to infinity)
Only absolutely convergent series are guaranteed to be rearrangement-invariant. ∑(−1)ⁿ/n² is absolutely convergent because ∑1/n² = π²/6 < ∞. Rearranging its terms always produces the same sum. By contrast, ∑(−1)ⁿ/n is only conditionally convergent (∑1/n diverges), so by Riemann's theorem it can be rearranged to any target value. The other two options diverge outright, so the question of rearrangement-invariance doesn't apply in the usual sense.
Question 3 True / False
If ∑aₙ converges absolutely, then ∑aₙ also converges in the ordinary sense.
TTrue
FFalse
Answer: True
Absolute convergence implies convergence. The proof uses the Cauchy criterion: since ∑|aₙ| converges, for any ε > 0 the tail sum ∑_{k=m}^{n} |aₖ| → 0, and since |∑aₖ| ≤ ∑|aₖ|, the partial sums of ∑aₙ form a Cauchy sequence and therefore converge. The converse is false — the alternating harmonic series converges but not absolutely.
Question 4 True / False
If ∑aₙ converges, then ∑|aₙ| also converges.
TTrue
FFalse
Answer: False
This is the classic false converse. The alternating harmonic series ∑(−1)ⁿ/n converges (to ln 2, by the alternating series test), but ∑|(−1)ⁿ/n| = ∑1/n is the harmonic series, which diverges. Convergence of ∑aₙ only implies absolute convergence when there is no cancellation structure exploiting the sign pattern — in general, alternating signs can carry a series to a finite limit even when the terms are too large for absolute convergence.
Question 5 Short Answer
Why does absolute convergence protect a series from the effects of rearrangement, while conditional convergence does not?
Think about your answer, then reveal below.
Model answer: In an absolutely convergent series, the positive and negative parts each converge independently to finite values. Rearranging terms cannot change the total because both sub-sums are locked in. In a conditionally convergent series, the positive terms alone diverge to +∞ and the negative terms alone diverge to −∞; the finite sum arises entirely from the balance between these two diverging components. By choosing a rearrangement that front-loads more positive (or more negative) terms, one can tip this balance arbitrarily, driving partial sums toward any target value or to infinity.
The Riemann rearrangement theorem is constructive: to rearrange ∑(−1)ⁿ/n toward a target T, take positive terms until the partial sum exceeds T, then take negative terms until it dips below T, and repeat. Since both the positive and negative sub-series diverge, this process never runs out of terms, and the oscillations shrink to zero because individual terms aₙ → 0. The key mechanism — inexhaustible supplies of both positive and negative terms — is exactly what absolute convergence rules out.