10.9 Determining Absolute or Conditional Convergence

Absolute vs. Conditional Convergence

A series Σaₙ converges absolutely if Σ|aₙ| converges. It converges conditionally if Σaₙ converges but Σ|aₙ| diverges. Absolute convergence is the stronger condition — it guarantees the original series converges too, since absolute convergence implies convergence.

Testing Strategy

To classify a series: first test Σ|aₙ| using tools like the Ratio Test, Comparison Test, or p-series rules. If |aₙ| converges → absolute convergence. If |aₙ| diverges but the original alternating series passes the Alternating Series Test, it's conditionally convergent.

Key Example: Alternating Harmonic Series

The series Σ(-1)ⁿ⁺¹/n converges by the Alternating Series Test, but Σ1/n (harmonic series) diverges. So this series is conditionally convergent — a classic example distinguishing conditional from absolute convergence.

Key terms

Absolute convergence

Σaₙ converges absolutely if Σ|aₙ| converges.

Conditional convergence

Σaₙ converges, but Σ|aₙ| diverges.

Key relationship

Absolute convergence implies convergence, but convergence doesn't imply absolute convergence.

Alternating harmonic series

Σ(-1)ⁿ⁺¹/n — classic example of conditional convergence (converges, but Σ1/n diverges).

Strategy for classification

Test Σ|aₙ| first (Ratio, Comparison, p-series); if it diverges, check original series with Alternating Series Test for conditional convergence.

Try an AP-style question

How does Σ (−1)ⁿ/n converge?

1. Absolutely
2. Conditionally
3. Geometrically
4. It diverges