Test Selection Flowchart

Real Analysis · Axiom Academy

Every series wears a clue. Learn to read the form of and reach for the one test that cracks it. One look at the form tells you which test to grab There is no single test that settles every series. Instead, each convergence test is tuned to a particular shape — a factorial, an alternating sign, a power of n , a familiar 1/n^p . The skill isn't memorizing a flowchart; it's learning to glance at , spot the feature that gives it away, and pick the test built for it. Watch the series ride in one at a time. For each, the feature that picks the test lights up, the matching test reaches over and claims it, and a verdict — converges or diverges — stamps onto the series. That feature-to-test match is the whole game. The feature in the series and the test that handles it are two halves of one move — read one, you know the other. Your turn: read the form, then pick the test Here's a series. Don't compute anything — just look at its structure and choose the test built for that shape. The feature that gives it away will light up, and you'll see the verdict it leads to. The form is the fingerprint: factorial → ratio, n th power → root, 1/n^p → p-series, alternating sign → alternating-series test. Wire each feature to the test it triggers Tap a feature on the left, then tap the test it should fire on the right. Correct pairs lock together with a line. Get all six and the map of "what selects what" is yours. When you can do this on sight, half of any convergence problem is already done.

This is the written version of the interactive lesson above. See the full Real Analysis course.