MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.6 – Alternating Series, Absolute and Conditional Convergence 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 The test tells you nothing! Each test has it limitations (i.e. conditions where the test fails).

NOTE: All un’s are assumed to be positive. Alternating Series OR NOTE: All un’s are assumed to be positive.

Alternating Series - Examples The alternating harmonic series (will prove to be convergent). An alternating geometric series (convergent because r = –1/2). A divergent alternating series (nth-term test).

The Alternating Series Test The series … Converges if …

The Alternating Series Test Converges if … Proof:

The Alternating Series Test Converges if … Proof:

The Alternating Series Test Converges if … Proof: Therefore, under these conditions, the alternating series converges.

Example 1 of the Alternating Series Test The Alternating Harmonic Series Decreasing? Limit? Therefore, convergent.

Example 2 of the Alternating Series Test Decreasing? Limit? Therefore, convergent.

Approximating Alternating Series If an alternating series satisfies the conditions of the alternating series test, and sn is used to approximate the sum; then … i.e. The error is less than the first term omitted.

Approximating Alternating Series Example: 1. Estimate the error if 4 terms are used to approximate the sum.

Approximating Alternating Series Example: 2. How many terms are need to make sure the error is less than 0.01? Therefore, four terms are needed!

Otherwise, it is not absolutely convergent. Absolute Convergence converges absolutely … i.e. is absolutely convergent if is convergent. Otherwise, it is not absolutely convergent. which does not mean that it is divergent

Absolute Convergence: Example 1 Convergent geometric series, therefore the first series converges absolutely. Note that the first series is NOT an alternating series.

Absolute Convergence: Example 2 Divergent harmonic series, therefore the first series is not absolutely convergent. NOTE: The first series IS a CONVERGENT alternating series.

Conditional Convergence If a convergent series is not absolutely convergent, it is said to be conditionally convergent. Example: Convergent alternating series. Divergent harmonic series. Therefore, the first series is “Conditionally Convergent.”

Absolute Convergence Test If a series converges absolutely, then it converges. convergent Proof: convergent Therefore, by the comparison test, the series converges.

Absolute Convergence Test: Example Convergent geometric series, therefore the first series converges absolutely. Therefore, the original series converges. NOTE: If it does not converge absolutely, the test fails!