Ratio and Root Tests

Real Analysis · Axiom Academy

LESSON The Ratio and Root Tests One verdict rule, two ways to find it: or — converge when the terms ultimately shrink geometrically. Form the ratio of each term to the one before it. If those ratios settle to a limit below 1 , then far out the series shrinks like a geometric series with common ratio L — so it converges absolutely. the verdict rule for the limit L Watch it on . The bars are the terms; the marker is their consecutive ratio sliding toward L on the scale below. 2. Where L Lands Decides the Verdict The single number L is read off a scale split at 1 . The same rule covers every series: the boundary L=1 is the one place the test stays silent — there the terms neither clearly shrink nor clearly grow, so you need a finer tool. Terms ultimately shrink geometrically. The series converges absolutely. The boundary. Inconclusive — the test gives no answer; try another. Terms ultimately grow, so they can't go to 0 . The series diverges . The marker visits three series in turn — watch how the bars answer to where L falls. When a term is itself an n th power, the cleaner limit is the n th root: . The same verdict rule applies — L<1 converges, L>1 diverges, L=1 is inconclusive — because again says the terms behave like . best for n th-power expressions — the root undoes the power On the n th root strips the exponent off in one move. Watch the root of each term climb toward its limit on the scale.

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