Convergence of a Telescoping Series

[Kulla_9289]

Hi all,
I have the following question. The first and second parts were proven. However, the third part does not make any sense to me or I do not understand. When x>1, each term of the telescoped sum grows without bound rather than shrinking to zero, so the partial sums cannot settle to a finite limit. This is how I understand it. Any help will be appreciated.

Thanks

 

[blamocur]

When x>1, each term of the telescoped sum grows without bound rather than shrinking to zero, so the partial sums cannot settle to a finite limit.

Agree.

This is how I understand it. Any help will be appreciated.

So what do you need help with?

 

[Kulla_9289]

The problem is that a solution has been given and I have been marked wrong.

 

[Dr.Peterson]

The problem is that a solution has been given and I have been marked wrong.

So what answer did you give that was marked wrong? And was anything specific said about what was wrong?

You haven't shown any answer to the question!

 

[Kulla_9289]

So what answer did you give that was marked wrong? And was anything specific said about what was wrong?

The answer given is "related to the condition of r in the sum of to infinity of a geometric sequence"

 

[Kulla_9289]

You haven't shown any answer to the question!

Because I do not know; ie. I don't even understand.

 

[Dr.Peterson]

Because I do not know; ie. I don't even understand.

I'm very confused. You said you gave a solution and it was marked wrong! So what was it?? Or did you mean the teacher gave a solution (what was that, then?), but your blank page was marked as wrong?

The question asked,

​

You seem to have implied a partial answer, namely that it converges when $x\le1$, by saying when it doesn't converge.:

When x>1, each term of the telescoped sum grows without bound rather than shrinking to zero, so the partial sums cannot settle to a finite limit.

That seems reasonable.

And your work for part b implies a way to find the sum to infinity:

​

What is the limit of that as x goes to infinity, if $x<1$, or if $x=1$?

Now you say,

The answer given is "related to the condition of r in the sum of to infinity of a geometric sequence"

That sounds more like a hint than an answer or a solution. Of course, this is not a geometric series (which is summed); but you have evidently given some thought to the behavior of a geometric sequence in what you have said.

In any case, you can get a much better answer if, rather than saying "I don't understand", you say something specific about what part of the problem you don't understand. (That's why I'm trying to be precise about what I don't understand about what you have said.)

 

[Kulla_9289]

"When x>1, each term of the telescoped sum grows without bound rather than shrinking to zero, so the partial sums cannot settle to a finite limit." This was marked wrong. The given answer is -1<x<1. As to what I do not understand, on how it's possible and how is it possible to figure this out?

 

[blamocur]

What is the limit of that as x goes to infinity,

I suspect you meant "as N goes to infinity..."

 

[Dr.Peterson]

I suspect you meant "as N goes to infinity..."

Yes, and I said at least one other thing that wasn't quite right ... my focus was on getting this person to say what they mean.

"When x>1, each term of the telescoped sum grows without bound rather than shrinking to zero, so the partial sums cannot settle to a finite limit." This was marked wrong. The given answer is -1<x<1. As to what I do not understand, on how it's possible and how is it possible to figure this out?

Ah, so you did in fact give an answer, and that is what was marked as wrong. That makes things a little clearer.

And the reply you were given was an explanation of how you can figure this out. (Though that, if you quoted it correctly, is a little off, because, as I suggested, this is not about the sum of a geometric sequence.)

Have you learned that a geometric sequence will converge when |r| <1? Do you see that x here plays the role of r?

But your mention of x>1 at least comes very close to this. How did you determine that? Do you not see how to change your answer to make it correct?

I think you need to have a chat with your teacher, to clarify what each of you means. Asking a third party doesn't help your communication with your teacher. (And if you want more help here, it might help to give a complete list of who said what to whom, so we can be sure which "answer" fits what question. I'm still not quite sure I understand the sequence of event. No pun intended.)