9-5 Alternating Series Rizzi – Calc BC

Objectives Use the Alternating Series Test to determine whether an infinite series converges. Use the Alternating Series Remainder to approximate the sum of an alternating series.

The simplest series that contain both positive and negative terms is an alternating series, whose terms alternate in sign. For example, the geometric series is an alternating geometric series with Alternating series occur in two ways: either the odd terms are negative or the even terms are negative. Alternating Series

Visual  An alternating series bounces back and forth between positive and negative values

Visual Part 2  Since the terms are always getting smaller, each added term gets us closer to approximating the actual sum:

Two Conditions for Convergence

Example 1  Does the following series converge or diverge?  Hint: since it’s alternating, check your two criteria

Example 2  Does the following series converge or diverge?

Absolute vs. Conditional Convergence  An alternating series converges absolutely if the absolute value of all of its terms converges  An alternating series converges conditionally if it converges, but not converges absolutely  Otherwise, the series diverges

Example: The Harmonic Series  Alternating Harmonic Series:

Error Bound  Since we sometimes want to sum a finite number of terms in an alternating series, we are interested in finding out the maximum error of our approximation  The error is the remainder (R n )after n terms

Error Bound Visual  Think back to this:  As the number of terms increases, our error is bounded to a smaller and smaller value

Error Bound Example  Consider the alternating harmonic series:  If we sum the first 6 terms, what is the maximum value of error for our approximation?

Number of Terms to Approximate  How many terms must be summed to approximate to three decimal places the value of the following series?

Homework  P. 625 #5-25, odd