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Infinite Series 9 Copyright © Cengage Learning. All rights reserved.
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Alternating Series Copyright © Cengage Learning. All rights reserved. 9.5
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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. Classify a convergent series as absolutely or conditionally convergent. Rearrange an infinite series to obtain a different sum. Objectives
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Alternating Series
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Example: This series is the alternating harmonic series – quite famous!
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Let’s look at some partial sums graphically. Clearly, the Partial Sums are headed for some limit in the middle of these points. Recall that if sequence of partial sums converges to S, then the series does too. Each “jump” is smaller than the previous one.
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However, all this is only true if each step (term) is smaller than the last step and if the steps (terms) are approaching zero. Each jump gets smaller, but never become less than ½ a unit. The jump up is always larger than the jump down, even though the sizes of the jumps in each direction are going towards zero.
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Alternating Series Test In other words… if a series alternates, checking to see if the terms are decreasing towards zero is sufficient.
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Ex: #1 This series converges by the Alternating Series Test.
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Ex: #2 The terms don’t appear to be decreasing in magnitude or going to 0 We need to look at some terms when n is large. The terms do eventually decrease in magnitude. This series converges by the A.S.T.
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Ex: #3 This series diverges by the A.S.T.
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Ex: #16 This series diverges by the nth term test.
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Ex: #4 You may want to quickly verify that the terms are decreasing. This series converges by the A.S.T.
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Ex: #6 This series diverges by the A.S.T. Or by nth term test
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Approximate the sum of the following series by its first six terms. Solution: The series converges by the Alternating Series Test because Example 6 – Approximating the Sum of an Alternating Series
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The sum of the first six terms is and, by the Alternating Series Remainder, you have So, the sum S lies between and you have Example 6 – Solution cont’d
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Absolute and Conditional Convergence
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Ex: #1 Determine the convergence or divergence of the series. If convergent, classify the type. What do you think?
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Determine whether each of the series is convergent or divergent. Classify any convergent series as absolutely or conditionally convergent. Example 2 – Absolute and Conditional Convergence
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a. By the nth-Term Test for Divergence, you can conclude that this series diverges. b. The given series can be shown to be convergent by the Alternating Series Test. Moreover, because the p-series diverges, the given series is conditionally convergent. Example 2 – Solution
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Ex: #3 Do the terms go to zero? This series converges by the A.S.T.
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Ex: #4 This series diverges by the nth Term Test. …and the crowd goes wild!
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Ex: #5 Estimate the sum of the infinite converging series: The error in an alternating series is less than the next term. So our answer is correct to the 7th decimal place.
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Rearrangement of Series
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A finite sum such as (1 + 3 – 2 + 5 – 4) can be rearranged without changing the value of the sum. This is not necessarily true of an infinite series—it depends on whether the series is absolutely convergent (every rearrangement has the same sum) or conditionally convergent. Rearrangement of Series
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The alternating harmonic series converges to ln 2. That is, Rearrange the series to produce a different sum. Solution: Consider the following rearrangement. Example 7 – Rearrangement of a Series
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By rearranging the terms, you obtain a sum that is half the original sum. Example 7 – Solution
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Homework Section 9.5 Day 1: pg.636: 11-29 odd, 35, 37, 47-59 odd, 79-84 all. Day 2: MMM 204-205
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