Absolute and Conditional Convergence

The central idea

A series $\sum a_n$ converges absolutely when $\sum|a_n|$ converges. Absolute convergence implies convergence of the original series. A convergent series is conditionally convergent when $\sum|a_n|$ diverges. The classification applies only after the original series has been shown to converge.

Definitions, hypotheses, and notation

The implication from absolute to ordinary convergence follows from comparison: $0\le |a_n|+a_n\le2|a_n|$, so positive and negative parts are controlled. The converse fails because cancellation can stabilize partial sums even when total absolute mass is infinite. A finite number of altered terms never changes either classification; convergence is determined by the tail.

When a power-series endpoint produces an alternating expression, this distinction is often the final step. One endpoint may converge absolutely, another conditionally, and another diverge, so the interval notation alone should be accompanied by endpoint classifications.

Conceptual meaning

Absolute convergence means signs are unnecessary for controlling the total size of the terms; conditional convergence depends on cancellation between positive and negative contributions. Absolute convergence permits rearrangement without changing the sum, while conditional rearrangements can behave very differently.

A dependable method and decision rule

1. Test $\sum|a_n|$ first with an appropriate positive-term test.
2. If the absolute series converges, stop and report absolute convergence.
3. If it diverges, return to $\sum a_n$ and test that signed series separately.
4. Use the alternating-series test only after checking decrease and limit zero.
5. Report conditional convergence only when the signed series converges but the absolute one does not.

Fully worked example

Common mistakes and why they fail