Each test is built for a particular kind of series. Here's what each one is really asking and when to reach for it — the "when" line is the trigger to memorize.
The cards above are for reading one at a time. This is for comparing them side by side — the single-screen reference to scan right before a test. Tap any column header to sort the focus, or just read across.
| Test | Use when… | The condition | What it proves | Watch out |
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Tip: the ratio & root tests are inconclusive when L = 1, and the nth-term test can only prove divergence — never convergence. Those two facts trip up the most people.
Once a series converges, there's a finer question the exam loves: does it still converge if you make every term positive? That split — absolute vs. conditional — is its own topic, and it hinges entirely on the alternating tests you just met.
The series of absolute values Σ|aₙ| converges too. The convergence is "robust" — it doesn't depend on cancellation between positive and negative terms.
The series converges as written, but Σ|aₙ| diverges. It only survives because of the alternating cancellation — remove the signs and it falls apart.
Why care? Absolute convergence is the stronger guarantee — you can rearrange the terms in any order and the sum won't change. Conditionally convergent series are fragile: rearranging them can actually change what they add up to.
Now the real skill: see a series, name the test you'd use. Instant feedback with the reasoning. This is what turns "I sort of remember the tests" into "I know which one on sight."