Unravelling the Mystery of Cn: How to Find Convergent Sequences

In summary, the conversation discusses finding convergent sequences related to a given sequence, Cn = [(-1)^n * 1/n!]. It is noted that this sequence does not converge to a single value, but by using the squeeze theorem, two sequences, An = -1/2^n and Bn = 1/2^n, can be found that converge to 0 and lie within the values of Cn. The use of the Absolute Value Theorem is also suggested as a way to find the convergence of sequences that jump back and forth.
  • #1
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0
Sequences HELP!

Homework Statement



Show that the sequence Cn = [(-1)^n * 1/n!]

Homework Equations


The Attempt at a Solution



This is an example in my book but I am not understanding it...

It says to find 2 convergent sequences that can be related to the given sequence. 2 possibilities are An = -1/2^n and Bn = 1/2^n
.....
where are they getting this from??
 
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  • #2
When trying to determine if a sequence converges it needs to approach a certain number. By pluging in numbers for the value of n you'll notice that

Cn = [(-1)^n * 1/n!]

will jump back and fourth between positive and negative values, and therefore isn't approaching a single value. You then need to find a graph both above and below that sequence that converge, you do this by the squeeze theorem.

If you graph all three of those sequences you'll notice that Cn = [(-1)^n * 1/n!] lies in-between. You can therefore say that because

1/2^n

and

-1/2^n

converge, the value in-between them converges also. What would that be the?
 
  • #3
but how do you come up with 1/2^n and -1/2^n?? The value would be 0?
 
  • #4
Yes all the values converge to zero. I really don't like to use the squeeze theorem, whenever I have to use it it's just by trial and error.

There should however be another theorem which relates to sequences that jump back and fourth. It's called the Absolute Value Theorem, which would allow you to more easily find the convergence of a sequence like this.
 
  • #5
ok thank you so much!
 

1. What is Cn and why is it important?

Cn, also known as the "convergence number," is a mathematical concept used to determine whether a sequence of numbers is convergent or divergent. It is an important tool in understanding the behavior of sequences and their limits.

2. How do you find convergent sequences?

To find convergent sequences, you must first examine the sequence and determine if it satisfies the definition of a convergent sequence. This involves checking if the terms in the sequence get closer and closer to a specific value, known as the limit. You can also use mathematical tests, such as the ratio test or the root test, to determine convergence.

3. What is the difference between a convergent and divergent sequence?

A convergent sequence has a limit that the terms in the sequence approach as the number of terms increases. In contrast, a divergent sequence does not have a limit and the terms in the sequence do not approach a specific value as the number of terms increases.

4. How does Cn relate to the concept of infinity?

Cn is related to infinity in that it helps us understand the behavior of sequences as the number of terms increases towards infinity. Convergent sequences approach a specific value as the number of terms increases, while divergent sequences do not. Cn is also used in determining the convergence or divergence of infinite series.

5. Can Cn be applied to real-life situations?

Yes, Cn can be applied to real-life situations in various fields such as physics, engineering, and economics. For example, it can be used to analyze the behavior of stock prices or the growth of bacteria in a petri dish. It is a valuable tool in understanding and predicting patterns and behaviors in the real world.

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