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17Calculus Infinite Series - Absolute and Conditional Convergence

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Absolute and conditional convergence applies to all series whether a series has all positive terms or some positive and some negative terms (but the series is not required to be alternating).

One unique thing about series with positive and negative terms (including alternating series) is the question of absolute or conditional convergence. Once convergence of the series is established, then determining the convergence of the absolute value of the series tells you whether it converges absolutely or conditionally. Formally, here's what it looks like.

Definitions of Absolute and Conditional Convergence

Given that \(\sum{a_n}\) is a convergent series.

if \(\sum{\left|a_n\right|}\) converges

 

if \(\sum{\left|a_n\right|}\) diverges

then \(\sum{a_n}\) converges absolutely

 

then \(\sum{a_n}\) converges conditionally

Here is a table that summarizes these ideas a little differently.

\(\sum{a_n}\)

\(\sum{\left|a_n\right|}\)

conclusion

converges

converges

\(\sum{a_n}\) converges absolutely

converges

diverges

\(\sum{a_n}\) converges conditionally

Absolute Convergence Theorem

Absolute Convergence Theorem

If the series \( \sum{\left|a_n\right|} \) converges,
then the series \( \sum{a_n} \) also converges.

Alternatively, it is possible to determine the convergence of the absolute value of the series first. Then, if the absolute value of the series converges, you can use the Absolute Convergence Theorem to say that the alternating series also converges and converges absolutely.

Additionally, if you have a series with some negative terms (but not all) and it is not an alternating series, you can use this theorem to determine convergence. Specifically, if the absolute value of the series converges, then the series will converge. Notice that this theorem says nothing about divergence, so you cannot make any assumption about convergence or divergence if this theorem does not hold.

Understanding The Theorem

Let's get an idea of why this theorem works. First, let's think about the series \( \sum{a_n} \) with positive and negative terms. As we add up the terms, we will be adding some positive terms and subtracting other terms (the negative ones). Compare that to the series \( \sum{\left|a_n\right|} \). With this series, we add all the terms as we go, i.e. we add all the positive terms and then we take the absolute value of the negative terms before we add those terms also. So each partial sum of this series is greater than the partial sum of the previous series. Consequently, as we continue to calculate the partial sums, we can say that the second sum is larger than the first sum. So if the larger sum converges, the smaller sum also has to converge.

Here is a simple example that should give you an intuitive feel for this.
For this finite alternating series \( 1 - 2 + 3 - 4 + 5 = 3 \).
Now calculate the absolute value of that series to get \( 1 + 2 + 3 + 4 + 5 = 15 \)
If we are always adding the numbers and never subtracting, the sum will always be larger for the absolute value series. And in the case of an infinite sum, if the larger series converges, logically, the smaller one will too.
Note - This is not a formal proof of the theorem. It is just an example to give you a feel for it.

Okay, let's watch some videos to get a better idea of what is going on. This next video clip has a great discussion on absolute convergence including using some examples. Notice that he does not use the term conditional convergence. Instead, he just says that the series does not converge absolutely.

Dr Chris Tisdell - Alternating series and absolute convergence [8min-44secs]

video by Dr Chris Tisdell

Here is another short video clip explanation of absolute and conditional convergence.

PatrickJMT - Absolute Convergence, Conditional Convergence and Divergence [2min-0secs]

video by PatrickJMT

Freaky Consequence of Conditionally Convergent Infinite Series

As you probably know by now, when you start working with numbers out to infinity, strange things happen. This is certainly true of infinite series which are conditionally convergent. The strange thing is that, when you rearrange the sum, you can get different values to which the series converges, i.e. the commutative property of numbers does not hold! Whoa!

In fact, not only can you get different values, it is possible to rearrange a conditionally convergent infinite series in order to get any number we want, including zero and infinity! This is called the Riemann Series Theorem.
Want to know more? Check out this wikipedia page.

Okay, time for some practice problems. These problems are not alternating series. For practice problems applying absolute and conditional convergence to alternating series, see the alternating series page.

Practice

Unless otherwise instructed, for each of the following series:
1. Determine whether it converges or diverges.
2. Determine if a convergent series converges absolutely or conditionally.

\(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{\sin(n)}{n^2} } }\)

Problem Statement

For the series \(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{\sin(n)}{n^2} } }\)
1. Determine whether the series converges or diverges.
2. Determine if a convergent series converges absolutely or conditionally.

Final Answer

The series converges absolutely by the direct comparison test and the absolute convergence theorem.

Problem Statement

For the series \(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{\sin(n)}{n^2} } }\)
1. Determine whether the series converges or diverges.
2. Determine if a convergent series converges absolutely or conditionally.

Solution

186 video

video by Dr Chris Tisdell

Final Answer

The series converges absolutely by the direct comparison test and the absolute convergence theorem.

close solution

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\( \displaystyle{\sum_{n=1}^{\infty}{ \frac{n^3+n}{n^5+4} } }\)

Problem Statement

For the series \( \displaystyle{\sum_{n=1}^{\infty}{ \frac{n^3+n}{n^5+4} } }\)
1. Determine whether the series converges or diverges.
2. Determine if a convergent series converges absolutely or conditionally.

Final Answer

The series converges absolutely.

Problem Statement

For the series \( \displaystyle{\sum_{n=1}^{\infty}{ \frac{n^3+n}{n^5+4} } }\)
1. Determine whether the series converges or diverges.
2. Determine if a convergent series converges absolutely or conditionally.

Solution

199 video

video by PatrickJMT

Final Answer

The series converges absolutely.

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Practice Instructions

Unless otherwise instructed, for each of the following series:
1. Determine whether it converges or diverges.
2. Determine if a convergent series converges absolutely or conditionally.

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