Root and Ratio Tests for Absolute Convergence

Real Analysis · Axiom Academy

LESSON Root and Ratio Tests for Absolute Convergence Both tests read — so delivers the strong conclusion: the series converges absolutely . Absolute convergence asks about — the series of magnitudes , with every sign stripped away. Both the ratio test and the root test put that absolute value in from the very first step, so whatever verdict they give is a verdict on . And converging is the strong outcome: it drags the original along with it. absolute convergence is the stronger conclusion — it implies convergence Take . Its terms alternate sign; watch them fold to the magnitudes that the tests actually weigh. 2. Why Gives Absolute Convergence Suppose the test returns . Pick any r strictly between L and 1 . From some point on, every magnitude obeys — the terms are trapped beneath a decaying geometric series . Since has a finite sum, the smaller does too. That comparison is the whole proof of absolute convergence. Here , so and . Watch a geometric cap with r=0.8 settle over the magnitude bars. 3. The Verdict — and the L=1 Trap Both tests report the same number L , read off the same scale split at 1 : means converges (absolute convergence), means the terms don't even reach 0 so the series diverges, and L=1 is the silent boundary — the test refuses to decide, and the series may go either way. is dominated by a geometric tail. Converges absolutely. The boundary. Inconclusive — no geometric room; use another test. Magnitudes grow, so . The series diverges .

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