Divergent alternating series problem

[Teachme]

Homework Statement

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 into 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?

Thanks for reading.

 

[micromass]

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 $\mathbb{R}$)

 

[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.

 

[micromass]

I suggest you look for a proof of this thing.

 

[Dick]

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.

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.

 

[Teachme]

Oh that makes sense. Thanks for the help.

 

[McAfee]

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