Absolute vs Conditional Convergence

A typical calculus student sees that a series like ∑(-1)ⁿ/n converges and assumes it is absolutely convergent. It is not. The series ∑1/n diverges (it is the harmonic series), so the alternating version only converges conditionally. The distinction between absolute vs conditional convergence is the difference between a series that is robust to rearrangement and one that is fragile. Every AP Calculus BC student and every Calculus II student must classify a series as absolutely convergent, conditionally convergent, or divergent. Getting this wrong means misapplying the Ratio Test or the Alternating Series Test and losing a full problem on the exam.

A series ∑a_n is absolutely convergent if ∑|a_n| converges. A series is conditionally convergent if ∑a_n converges but ∑|a_n| diverges. A series that is not convergent is divergent. This is the vocabulary from Stewart, Calculus, section 11.6 and AP Calculus BC CED Topic 10.9 (2024). Only an alternating series can be conditionally convergent. A series of positive terms that converges is automatically absolutely convergent.

If a series is absolutely convergent, it is convergent. This theorem from Stewart, section 11.6 means you never need to check the signed series separately once you have proven absolute convergence. The converse is false: a convergent series may or may not be absolutely convergent. The alternating harmonic series ∑(-1)ⁿ⁺¹/n converges to ln 2 (Stewart, chapter 11) but its absolute series ∑1/n diverges. So it is conditionally convergent, not absolutely convergent.

Step 1: Test the series of absolute values ∑|a_n| using the Ratio Test, Root Test, Comparison Test, Limit Comparison Test, Integral Test, p-Series Test, or Geometric Series Test. If ∑|a_n| converges, the original series is absolutely convergent. Step 2: If ∑|a_n| diverges, check whether the original series ∑a_n is alternating. If it is not alternating and ∑|a_n| diverges, the original series diverges. Step 3: For an alternating series that passes the Alternating Series Test (terms decrease to 0), the series is conditionally convergent. If the Alternating Series Test fails, the series diverges.

Start at ∑a_n. Compute ∑|a_n|. If that converges, label the series absolutely convergent. If ∑|a_n| diverges, ask: is the series alternating? If no, the series diverges. If yes, apply the Alternating Series Test. If the alternating terms decrease to 0, the series is conditionally convergent. If not, the series diverges. This procedure is the standard strategy from Stewart, section 11.7 and Paul's Online Math Notes Strategy for Series.

Example 1: ∑(-1)ⁿ/n²

∑|(-1)ⁿ/n²| = ∑1/n². This is a p-series with p=2, which converges (Stewart, section 11.3). Therefore the original series is absolutely convergent. No further testing needed. The sum is π²/6 (Basel problem, Euler, 1734).

Example 2: ∑(-1)ⁿ/n

∑1/n is the harmonic series, which diverges. The original series is alternating. Check the Alternating Series Test: b_n=1/n decreases to 0. So the series converges conditionally. Its sum is -ln 2 (Stewart, chapter 11).

Example 3: ∑(-1)ⁿn/(n+1)

∑|n/(n+1)| diverges because the terms do not approach 0. The absolute series fails the Divergence Test (Stewart, section 11.2). The original series also fails the Divergence Test: lim a_n ≠ 0. So the series diverges.

An inconclusive result on the Ratio Test or Root Test occurs when the limit L=1. For example, the Ratio Test on ∑1/n² gives L=1 and tells you nothing. This does not mean the series diverges. It means the test cannot decide. For L=1 cases, switch to a different test. Try the Comparison Test, Limit Comparison Test, or p-Series Test. For ∑1/n², the p-Series Test (Stewart, section 11.3) immediately shows convergence because p=2>1. For ∑1/n, the Ratio Test is also inconclusive, but the p-Series Test shows divergence because p=1. If you see an inconclusive result, do not stop. Apply the test from the next tier in the Strategy for Series (Paul's Online Math Notes).

When the Ratio Test or Root Test gives L=1, the test is inconclusive. You must use the Integral Test, Comparison Test, Limit Comparison Test, or a known test like the p-Series Test or Geometric Series Test. For example, the Ratio Test on ∑1/n³ gives L=1. Use the p-Series Test: p=3>1, so the series converges. Never claim convergence or divergence from an inconclusive test. The mistake of assuming L=1 means divergence costs students points on AP Calculus BC exams.

A conditionally convergent series can be rearranged to sum to any real number, including infinity. The Riemann rearrangement theorem (Stewart, section 11.6) states that if ∑a_n converges conditionally, then for any real number S, there exists a rearrangement that sums to S. Absolutely convergent series are safe: any rearrangement converges to the same sum. This is why absolute convergence is the stronger property. For example, the alternating harmonic series can be rearranged to sum to π or 0. On the AP exam, you only need to know the existence of the theorem, not its proof.