The Ratio Test

Calculus 2 · Axiom Academy

One limit decides everything: compare each term to the one before it and watch the series converge, diverge, or sit on the fence. 1. The Ratio Test: One Limit, One Verdict For a series , form the ratio of each term to the one before it, , and take the limit. That single value L tells you whether the terms are shrinking fast enough for the sum to settle down. the long-run ratio of consecutive terms If , the series converges absolutely If L = 1 , the test is inconclusive — try another test 2. The Three Cases on the Number Line The whole test lives on one axis. Mark L on the number line: which side of 1 it lands on is the verdict. Only the exact value L = 1 is undecided. 3. A Factorial Example, Worked The Ratio Test shines on factorials and exponentials, where the ratio collapses cleanly. Take this series — watch the ratio , built straight from a_n , climb without bound. 4. The Payoff: Radius of Convergence Apply the Ratio Test to a power series and something beautiful happens: the variable x comes along for the ride. Convergence ( ) becomes a condition on |x| — and that carves out an interval. The ratio limit is below 1 , so the power series converges. The ratio limit exceeds 1 , so the power series diverges. You've seen how one limit — the ratio of consecutive terms — decides convergence, and how that same idea pins down the radius of a power series. Scroll up to revisit any step.

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