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D. 8.5 Alternating Series

The document discusses the Alternating Series Test, which states that a series converges if its terms alternate in sign and decrease in magnitude to zero. It also covers the concept of the Alternating Series Remainder, explaining how to estimate truncation error and determine convergence or divergence of series, including absolute and conditional convergence. Several examples are provided to illustrate these concepts and methods for analyzing series.

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27liangh
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53 views15 pages

D. 8.5 Alternating Series

The document discusses the Alternating Series Test, which states that a series converges if its terms alternate in sign and decrease in magnitude to zero. It also covers the concept of the Alternating Series Remainder, explaining how to estimate truncation error and determine convergence or divergence of series, including absolute and conditional convergence. Several examples are provided to illustrate these concepts and methods for analyzing series.

Uploaded by

27liangh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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March 7, 2025

8.5: Alternating Series


The Alternating Series Test

converges if

In other words, this means that if the terms of a series are alternating in sign
and decreasing in magnitude to zero, then the series converges.
*Careful! If those things aren't true, the series doesn't necessarily diverge!
You just can't use the Alternating Series Test!
This test is used to prove convergence only!

*The test fails when one or both of these conditions are not met!
(If that happens, try the nth Term Test!)
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Be Careful! Alternating series can come in many different forms!


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Determine whether the following series converge or diverge.

Example 1:
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Example 2:
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Example 3:
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Example 4:
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Example 5:
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The Alternating Series Remainder


For a partial sum of an alternating series, you can actually find the truncation
error without having to use other (slightly more complex) methods, which we
will learn later.
Let's look back at the Alternating Harmonic Series to see what's going on.

Here are the first partial sums for this series:

You can see that they are heading to some value (the infinite sum, because this
series converges) and with each term added in, the next point doesn't go any
farther away from the final "sum" than the previous one.
Therefore, the amount of error (from the actual infinite sum) has to be less than
the magnitude of the first missing term! So if we use , that value
(the partial sum) is off from the actual sum of the series by less than .
error < |first missing term|
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Example 6: Approximate the sum of the series by its first 6 terms.


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Example 7: Determine the number of terms required to approximate the sum


of the series with an error of less than 0.001.
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Absolute and Conditional Convergence


1. is absolutely convergent if converges

2. is conditionally convergent if converges but diverges

If converges, then is said to converge absolutely and converges.

The converse is NOT true:


If converges then doesn't necessarily converge.
If converges but doesn't, then converges conditionally.

To determine if a series converges absolutely or conditionally, check to see if


converges. If it does, then converges absolutely so converges.

If diverges, then check . If it converges, then converges


conditionally. If it diverges, then diverges.

*Remember, if you can identify that the original series diverges, then you're done!
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Determine the convergence or divergence of the series. Classify any


convergent series as absolutely or conditionally convergent. Show the work
that leads to your conclusion.

Example 8:

*Note: this is not an alternating series, nor is it geometric!

Think of it this way... if you make all the terms positive and the series
converges, then if some of the terms are actually negative, it still has to
converge and will just converge to a smaller sum than the original.
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Example 9:
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Example 10:

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