# Conditional Convergence

## Main Question or Discussion Point

So we learned about the basic tests for convergence of an infinite series, and we learned about alternating series, and conditional convergence.

Now, I get how to find if a series is conditionally convergent. But what's the use of conditionally convergent infinite series? All we were taught was how to determine if it was one, but not any of the uses of this particular type of series, could anyone elaborate?

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daniel_i_l
Gold Member
Well one cool theorem (by Riemann) about a conditionally convergent infinite series is that by changing the order of the elements in the series you can make it converge to any number - rational or irrational.

HallsofIvy
Homework Helper
What do you mean by "use"?

I mean application of it.

Ok I have a conditionally convergent series, so what? Why is this important?

Hurkyl
Staff Emeritus
Gold Member
It's a classification method -- a convergent series does so either absolutely or conditionally. A relevant one too: many theorems about series behaving nicely only apply to absolutely convergent series, and many theorems about series behaving badly only apply to conditionally convergent series.

Yeah I'm starting to realize that... Could you list some of those theorems? I'd like to try and learn them :)

Hurkyl
Staff Emeritus
Gold Member
The most prominent has to do with rearranging series. When all the relevant series converge absolutely, you can rearrange the terms in any way you please without changing the answer. And daniel_i_l already stated the complementary fact about conditionally converging sequences.

arildno
Homework Helper
Gold Member
Dearly Missed
I mean application of it.

Ok I have a conditionally convergent series, so what? Why is this important?
Sigh.
Why is it important to classify an apple as a round fruit and a banana as a long fruit?

Can you give me an application of that knowledge, please?

arildno
Homework Helper
Gold Member
Dearly Missed
I mean application of it.

Ok I have a conditionally convergent series, so what? Why is this important?
Sigh.
Why is it important to classify an apple as a round fruit and a banana as a long fruit?
Do you understand what would happen to you if you mixed up those descriptions in public?

Furthermore:
Can you give me an application of that knowledge, please?

Sigh.
Why is it important to classify an apple as a round fruit and a banana as a long fruit?
Do you understand what would happen to you if you mixed up those descriptions in public?

Furthermore:
Can you give me an application of that knowledge, please?
I cannot give you application to that knowledge. However what would be the point in even mentioning conditionally convergent series if there was not some further application or usefulness to them? I don't see at what you're trying to get?

The most prominent has to do with rearranging series. When all the relevant series converge absolutely, you can rearrange the terms in any way you please without changing the answer. And daniel_i_l already stated the complementary fact about conditionally converging sequences.
As I see, that seems like a pretty good reason to learn about conditional convergence.

arildno