9.6 The Ratio & Root Tests Objectives: Use the Ratio Test for series convergence Use the Root Test for series convergence Review tests for convergence and divergence of infinite series ©2004Roy L. Gover (www.mrgover.com)

Important Idea Theorem-Ratio Test ( test for absolute convergence) in 3 parts: Let have non-zero terms: 1. converges absolutely if

Important Idea Theorem-Ratio Test ( test for absolute convergence) 2. diverges if or

Important Idea Theorem-Ratio Test ( test for absolute convergence) 3. The Ratio Test is inconclusive if The Ratio Test works best with series involving factorials or exponentials

Example Determine the convergence or divergence of: P590, ex1

Try This Determine the convergence or divergence of: Diverges P590, Ex 2b Diverges

Try This Determine the convergence or divergence using the Ratio Test: P591,ex 3 Inconclusive

Important Idea Theorem-The Root Test (in 3 parts): Let have non-zero terms: 1. converges absolutely if

Important Idea Theorem-The Root Test (in 3 parts): 2. diverges if or

Important Idea Theorem-The Root Test (in 3 parts): 3. The Root Test is inconclusive if The Root Test works best with series involving nth powers

Example Determine the convergence or divergence of: P592 ,ex 4-converges

with Mrs. DeBelina There are 10 tests for convergence and divergence of a series. So many tests; so little time. How do I decide?

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 1. nth Term Test for divergence: divergence and you are done.

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 2. Determine if the series is one of the special types: a) geometric b) p c) harmonic d) telescoping e) alternating

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: a) geometric series converges to if otherwise diverges.

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: b) p series converges if p>1 ,otherwise diverges

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: c) harmonic series is a special case of the p series where p=1. It diverges. The general form is

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: d) telescoping series converges to b1-L if

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: e) alternating series converges if and

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 3. Direct Comparison with a geometric or p-series known to converge. If and converges, then converges.

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 4. Limit Comparison with a geometric or p-series known to converge. If and converges, then converges.

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 5. Ratio Test for exponentials or factorials. converges if L<1 and diverges if L>1 where

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 6. Root Test for nth powers. converges if L<1 and diverges if L>1 where

Important Idea Applying the tests in this order, as applicable, means you don’t duplicate work: 7. Integral Test (f must be continuous, positive and decreasing). and both converge or diverge.

Lesson Close A summary of the convergence/divergence tests for infinite series is found on page 644 of your text.