The Convergence Test Toolkit

Calculus 2 · Axiom Academy

A whole shelf of tests for infinite series — the real skill is reading a series' shape and reaching for the right one. One series, six tests, and the question of which one Once a sum runs to infinity, you can't just add it up — you ask a yes/no question instead: does it converge? Calculus hands you a whole shelf of tests to answer it. The trap is that they all "work" on the right series and waste your time on the wrong one, so the real skill isn't memorizing six rules — it's reading a series' shape and reaching for the tool that fits. Watch one series get routed. We drop in at the top and let it fall past each test in order. The early gates don't fit its shape, so they wave it through; the factorial and the exponential snap it into the Ratio Test , which fires and returns a verdict. The series never changed — the toolkit just funneled it to the one test built for its shape. Each series below wears a tell. Click one and the toolkit answers: which test fits, the feature in the series that gives it away, and the verdict. Notice you're not computing anything yet — you're pattern-matching, which is most of the battle. Factorials whisper "Ratio." An power whispers "Root." A power in the denominator whispers " p -series." Learn the whispers. What "the Ratio Test fires" actually means

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