Finding a convergent subsequence of the given sequence

In summary, the problem is that the sequence won't converge because it oscillates between positive and negative numbers.
  • #1
Maddiefayee
5
0

Homework Statement


For some background, this is an advanced calculus 1 course. This was an assignment from a quiz back early in the semester. Any hints or a similar problem to guide me through this is greatly appreciated! Here is the problem:
Find a convergent subsequence of the sequence:
{(-1)n (1-(1/n)}n=1

Homework Equations


I don't think there are any equations needed. The class is all about proofs. Here's a definition:
A sequence {an} is said to converge to the number a provided that for every positive number ε there is an index N such that:
|an - a| < ε , for all indices of n ≥ N

The Attempt at a Solution


So this was my attempt. My "solution" was: {(1/n2)(-1)n}10n=2
How I got to this solution was honestly listing out a few terms of the original sequence and then finding another sequence that I thought would make sense.

Here's the note from my professor: "Sequences, and subsequences, have an infinite number of terms. Also, this sequence is not a subsequence."
 
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  • #2
Maddiefayee said:
Here's the note from my professor:
So do you now understand what constitutes a subsequence and why your attempt was not one?
 
  • #3
haruspex said:
So do you now understand what constitutes a subsequence and why your attempt was not one?
Yes! I understand that it is not a subsequence. I think I am just overthinking the problem here.
 
  • #4
Maddiefayee said:
Yes! I understand that it is not a subsequence.
Good.

So, write out four or five terms of the full sequence. What is it about the full sequence that means it will not converge?
 
  • #5
It won't converge because it oscillates between positive when n is even and negative when n is odd.
 
  • #6
Maddiefayee said:
It won't converge because it oscillates between positive when n is even and negative when n is odd.
Well, that's not enough in itself to prevent convergence. (-1)n/n converges happily. But you are right that it is part of the problem here. So how can you select a subsequence to avoid it?
 
  • #7
I could make it so that n=2k where k is in ℝ.
 
  • #8
Maddiefayee said:
where k is in ℝ.
You mean ℕ, right?
What would that give? Can you show it converges? (Or maybe you are not required to prove it.)
 
  • #9
Right. So then the new sequence would be:

{(-1)2n(1-(1/2n))}n=1

It would converge to 1.
 
  • #10
Maddiefayee said:
Right. So then the new sequence would be:

{(-1)2n(1-(1/2n))}n=1

It would converge to 1.
Right, but please use parentheses correctly: 1-1/(2n).
 

Related to Finding a convergent subsequence of the given sequence

1. What is a convergent subsequence?

A convergent subsequence is a sequence that is a subset of the original sequence and approaches a specific limit as the number of terms in the subsequence increases.

2. How do you find a convergent subsequence?

To find a convergent subsequence, you need to first identify a subsequence that is bounded and monotonic. Then, you can use the Bolzano-Weierstrass theorem to show that the subsequence converges to a limit.

3. What is the significance of finding a convergent subsequence?

Finding a convergent subsequence allows us to determine the limit of a sequence, which can provide important information about the behavior and properties of the original sequence. It also helps us to better understand the convergence or divergence of a sequence.

4. Can a sequence have more than one convergent subsequence?

Yes, a sequence can have multiple convergent subsequences. In fact, a sequence can have infinitely many convergent subsequences, each approaching a different limit.

5. Are there any other methods for finding a convergent subsequence?

Yes, there are other methods such as the Cauchy principle and the Cesaro mean, which can also be used to find convergent subsequences. However, the Bolzano-Weierstrass theorem is often the most commonly used method.

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