MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.5 – The Ratio and Root Tests Copyright © 2009 by Ron Wallace, all rights reserved.

Does the Series Converge? 10 Tests for Convergence Geometric Series N-th Term Test (Divergence Test) Integral Test p-Series Test Comparison Test Limit Comparison Test Ratio Test Root Test Alternating Series Test Absolute Convergence Test Each test has it limitations (i.e. conditions where the test fails). The test tells you nothing!

The Ratio & Root Tests NOTE: For all series in this section, it will be assumed that each term is positive. That is, given then

The Ratio Test Geometric Series! Convergent if |r|<1 Comparison Test  If L < 1, the series converges. (from the limit definition)

The Ratio Test Geometric Series! Divergent if |s|>1 Comparison Test  If L > 1, the series diverges. (from the limit definition)

The Ratio Test If L=1, the test fails!

The Ratio Test If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test fails.

Example w/ the Ratio Test  Divergent!

The Root Test If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test fails. Proof is similar to the ratio test!

Example w/ the Root Test  Convergent!