# Divergent alternating series problem

1. Dec 16, 2011

### Teachme

1. The problem statement, all variables and given/known data

If Ʃa_n is divergent, the absolute value of Ʃa_n is divergent. True or false. This is the main question I am trying to answer. I should be able to answer this problem on my own, but i ran into a problem that confused me.

What I Did

So I decided to start this question off by looking at some alternating series. I figured this would be the most likely to make this statement FALSE.
So I started off by looking at a divergent alternating series on my way I ran into a little problem that confused me. I attached a photo of a solution to a divergent alternating series that I found confusing. ( ). As you can see the test diverges. And this is my problem. When I plugged this in to my maple and approximated it went to -.006.... I don't understand why this would happen if the series is divergent. I included a picture of my approximation on maple. ( ) I might be wrong, but wouldn't you expect to get nothing if the series was divergent?

2. Dec 16, 2011

### micromass

Staff Emeritus
I can't explain Maple's answer. I guess that it is because maple cannot evaluate an infinite number of terms. It evaluates a finite number and then displays the result.

You are indeed correct that the series you suggest doesn't converge. Mainly because the limit

$$\lim_{n\rightarrow +\infty}{ \frac{(-1)^n n^2}{n^2+5}}$$

doesn't exist.

However, the absolute value of the series also doesn't converge.

(also do you mean to say that $\sum |a_n|$ must diverge?)

(and also, are you only working with series in $\mathbb{R}$)

3. Dec 16, 2011

### Teachme

Yeah that is what I was saying. Sorry I didn't know how to find an absolute value sign. I guess that initial question is sort of trivial, but I was just curious and was thinking of all sorts of examples that could lead to that statement being false.

The thing about maple though is that when I evaluate any other divergent series with n=1.. infinity.... Say I input Ʃan and then I got to approximate then the output is the same as my input Ʃan. So I have always assumed that when that happens it shows the series is just divergent so seeing this really threw me off and I don't have an explanation for it. Its obviously a fault with the program, I just can't see what it is.

Thanks a lot for the help.

Sorry for the sort post before, not abiding by the rules. Was just being ignorant not reading them to begin with.

4. Dec 16, 2011

### micromass

Staff Emeritus
I suggest you look for a proof of this thing.

5. Dec 16, 2011

### Dick

I'm guessing that what Maple is doing is regrouping your series into (n+1)^2/((n+1)^2+5)-n^2/(n^2+5) and then summing that over n=1,3,5,7,... That series does converge. Regrouping a divergent series to a convergent one is possible but it's illegal and misleading. Another reason to distrust Maple on questions like this.

6. Dec 16, 2011

### Teachme

Oh that makes sense. Thanks for the help.

7. Dec 17, 2011

### McAfee

Make sure you know the rules and expectations for the test.