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

Graphical or geometric meaning

Partial sums alternate above and below their limit, with shrinking corrections. Removing the signs turns every correction upward and produces unbounded harmonic growth. The contrast makes cancellation, rather than small terms alone, visible.

Common mistakes and why they fail

Verification and reasonableness checks

• Write the absolute-value series explicitly.
• Verify $a_n\to0$ before any convergence claim.
• Name separate tests and conclusions for the absolute and signed series.

Use absolute convergence as the stronger test

Begin with $\sum |a_n|$ because its convergence immediately proves convergence of $\sum a_n$. If the absolute-value series diverges, the original series is not yet classified; cancellation may still give conditional convergence. This order prevents the false inference that failure of absolute convergence means divergence. Conditional convergence also signals sensitivity to rearrangement, whereas absolutely convergent series may be rearranged safely. When classification is requested, state both conclusions explicitly: identify a test proving the original series converges and a test showing the absolute-value series diverges. Checking $a_n\to0$ remains a necessary preliminary, but it cannot establish either form of convergence. Preserve eventual positivity and other hypotheses when choosing each comparison or alternating-series argument.

Practice

1. Classify $\sum(-1)^n/n^2$.
2. Classify $\sum(-1)^{n+1}/n$.
3. Can an absolutely convergent series diverge?

Connections and next steps

• Alternating Series Test
• Comparison Test
• Infinite Series