# 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.

## 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."

Here's the note from my professor:
So do you now understand what constitutes a subsequence and why your attempt was not one?

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.

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?

It won't converge because it oscillates between positive when n is even and negative when n is odd.

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?

I could make it so that n=2k where k is in ℝ.

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.)

Right. So then the new sequence would be:

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

It would converge to 1.

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).

## 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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