AP Calculus BC
Absolute vs Conditional Convergence
A series converges absolutely when the series of absolute values also converges. It converges conditionally when it converges as written but the absolute values diverge, meaning the convergence depends entirely on cancellation between positive and negative terms.
Absolute convergence
Use when: The series of absolute values converges, which also proves the original converges.
Conditional convergence
Use when: The alternating series test passes but the absolute values form a divergent series.
Side by side
| Absolute | Conditional | |
|---|---|---|
| Original series | Converges | Converges |
| Series of absolute values | Converges | Diverges |
| Depends on cancellation | No | Yes |
| Example |
Deciding takes two tests, not one. Apply the alternating series test to the original, then test the absolute values separately. Convergent plus divergent means conditional; convergent plus convergent means absolute.
The ratio and root tests are stated with absolute values, so whenever they conclude convergence they have already proved absolute convergence and no second test is needed.
Why absolute is stronger
Absolute convergence always implies convergence, so establishing it lets you use tests that require positive terms on a series with mixed signs.
Frequently asked questions
Can a series of positive terms converge conditionally?
No. With no negative terms the series and its absolute values are identical, so any convergence is absolute.
In the CED: Unit 10: Infinite Sequences and Series (BC)