Calculus/Absolute and Conditional Convergence

Absolute and Conditional Convergence
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
Here is a table that summarizes these ideas a little differently.

One minor point is that all positive series converge absolutely since $$ a_n = |a_n| $$ for all $$ n $$.

Absolute Convergence Theorem
This theorem uses the first row of the above table and this allows us to consider the divergence of a positive series.

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.

Video Recommendations
This 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.

Alternating series and absolute convergence

Here is another short video clip [2min] explanation of absolute and conditional convergence.

Absolute Convergence, Conditional Convergence and Divergence

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