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F) The Alternating Series Test (AST) Theorem 6

This document discusses alternating series and different types of convergence for series. It introduces the Alternating Series Test, which provides criteria for determining if an alternating series converges or diverges. Specifically, an alternating series converges if the terms decrease to zero. It also discusses absolute convergence, where a series converges if the corresponding series of absolute values converges, and conditional convergence, where a series converges but is not absolutely convergent. A flow chart is provided that outlines the steps to check the convergence of different types of series.

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Syahmi
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85 views3 pages

F) The Alternating Series Test (AST) Theorem 6

This document discusses alternating series and different types of convergence for series. It introduces the Alternating Series Test, which provides criteria for determining if an alternating series converges or diverges. Specifically, an alternating series converges if the terms decrease to zero. It also discusses absolute convergence, where a series converges if the corresponding series of absolute values converges, and conditional convergence, where a series converges but is not absolutely convergent. A flow chart is provided that outlines the steps to check the convergence of different types of series.

Uploaded by

Syahmi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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CHAPTER1 MAT455

1.4 Alternating series

The convergence tests that we have looked at so far apply only to series with positive
terms. An alternating series is a series whose terms are not all positive, but alternate in
sign (alternately positive and negative).

F) The Alternating Series Test (AST)

Theorem 6
The series of the form

∑∞ 𝑛
𝑛=1(−1) 𝑎𝑛 = −𝑎1 + 𝑎2 − 𝑎3 + 𝑎4 … … … .

or ∑∞
𝑛=1(−1)
𝑛+1
𝑎𝑛 = 𝑎1 − 𝑎2 + 𝑎3 − 𝑎4 … … … .

converges if i) an+1 ≤ an for all n ( the sequence an is decreasing)


ii) lim𝑛→∞ 𝑎𝑛 = 0.

Otherwise, the series diverges.

Example 1
Determine if the following series converge or diverge.

(−1)𝑛−1 (−1)𝑛 3𝑛 (−1)𝑛+1 𝑛2


a) ∑∞
𝑛=1 b) ∑∞
𝑛=1 c) ∑∞
𝑛=1
𝑛 4𝑛−1 𝑛3 +1

1.4.1 Absolute Convergence

We have convergence tests for series with positive terms and for alternating series.
But what if the signs of the terms switch back and forth irregularly? We will see that
the idea of absolute convergence sometimes helps in such cases.

Given any series ∑ 𝑎𝑛 , we can consider the corresponding series

∑∞
𝑛=1|𝑎𝑛 | = |𝑎1 | + |𝑎2 | + |𝑎3 | + |𝑎4 | +…….

whose terms are the absolute values of the terms of the original series.

Definition 5

A series ∑ 𝑎𝑛 is called absolutely convergent if the series of absolute values ∑|𝑎𝑛 | is


convergent.

Note: if ∑ 𝑎𝑛 is a series with positive terms, then |𝑎𝑛 | = 𝑎𝑛 and so absolute


convergence is the same as convergence in this case.

sumarni Page 10
CHAPTER1 MAT455

Example 2
The series
(−1)𝑛−1 1 1 1
∑∞
𝑛=1 = 1 − 22 + 32 − 42 +………..
𝑛2

is absolutely convergent because

(−1)𝑛−1 1 1 1 1
∑∞
𝑛=1 | | = ∑∞
𝑛=1 𝑛2 = 1 + + 32 + 42 … … …
𝑛2 22

is a convergent p-series (p = 2).

Definition 6
A series ∑ 𝑎𝑛 is called conditionally convergent if it is convergent but not absolutely
convergent.

Example 3
We know that the alternating harmonic series

(−1)𝑛−1 1 1 1
∑∞
𝑛=1 =1−2+ − ………
𝑛 3 4

is convergent (as discussed earlier), but it is not absolutely convergent because the
corresponding series of absolute values is

(−1)𝑛−1 1 1 1 1
∑∞
𝑛=1 | | = ∑∞
𝑛=1 𝑛 = 1 + 2 + + ………
𝑛 3 4
is a divergent harmonic series.
(apply AST to check for conditional convergence)

Example 4
Determine whether the following series is absolutely convergent, conditionally
convergent, or divergent.

(−1)𝑛 𝑛3 (−1)𝑛+1 (−1)𝑛 𝑛


a) ∑∞
𝑛=1 b) ∑∞
𝑛=1 4 c) ∑∞
𝑛=1
3𝑛 √𝑛 5+𝑛

TRY THIS

(−1)𝑘 𝑘
Determine whether the series ∑∞
𝑘=1 𝑘 2𝜋 +4 converges absolutely, converges
conditionally or diverges.

(Ans: absolute convergence)

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CHAPTER1 MAT455

A FLOW CHART OF STEPS TO CHECK THE CONVERGENCE OF SERIES

Regular/Irregular alternate series


or positive series
(original series)

Consider
∑∞
𝑛=1|𝑎𝑛 |

Test the convergence


of the series

converges diverges

∑∞
𝑛=1|𝑎𝑛 | converges, thus
Check using AST for Satisfy both
original series converges conditionally convergence conditions
absolutely

Not satisfy both Original series


conditions converges
conditionally

Original series
diverges

sumarni Page 12

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