Convergence/Divergence of this sequence

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In summary, the conversation discusses a sequence and its convergence or divergence. One person provides a solution based on intuition, but another suggests subtracting one term from the next to get a sum of positive terms. The conversation concludes that the sequence diverges based on the limits of the terms.
  • #1
fishingspree2
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Hello,

I am trying to find if this sequence is converging or diverging:

http://img9.imageshack.us/img9/623/72142860.gif

Here is my work, but I highly doubt it because it is based on intuition:
http://img4.imageshack.us/img4/3589/54428513.gif
Since the first limit diverges and the second limit is not equal to 0, then the sequence diverges.
Does anyone have a better, systematic way of dealing with this sequence and any other sign alternating sequence with (-1)^n

Thank you
 
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  • #2
Hello fishingspree2! :smile:
fishingspree2 said:
Hello,

I am trying to find if this sequence is converging or diverging:

http://img9.imageshack.us/img9/623/72142860.gif
[/URL]

Do you mean the sum, ∑ of that?

Hint: try subtracting one term from the next … that'll give you a sum of positive terms. :wink:
 
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  • #4
oooh sorry :redface: …

i was confused by the square brackets

(i wish the latex was back :frown:)

in that case …
fishingspree2 said:
Here is my work, but I highly doubt it because it is based on intuition:
http://img4.imageshack.us/img4/3589/54428513.gif
Since the first limit diverges and the second limit is not equal to 0, then the sequence diverges.

Why do you doubt it … that's fine! :biggrin:

Just tune it slightly by saying that the even terms converge to 2, and the odd terms converge to -2, so clearly it has no limit. :smile:
 
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  • #5
thank you tiny-tim :)
 

1. What is meant by "convergence" and "divergence" of a sequence?

"Convergence" and "divergence" refer to the behavior of a sequence as its terms continue to increase. A sequence is said to converge if its terms approach a specific limit or value as the sequence goes on. On the other hand, a sequence is said to diverge if its terms do not approach a limit and instead grow infinitely.

2. How can I determine if a sequence converges or diverges?

There are various methods for determining the convergence or divergence of a sequence, such as the limit comparison test, ratio test, and root test. However, the most common method is the basic comparison test, where the terms of the sequence are compared to a known convergent or divergent sequence. If the terms are smaller or larger than the known sequence, then the original sequence will have a similar behavior.

3. What is the importance of knowing if a sequence converges or diverges?

Knowing the convergence or divergence of a sequence is crucial in many mathematical and scientific applications. For instance, in calculus, the convergence of a sequence is related to the convergence of a corresponding series. In physics, the convergence or divergence of a sequence can determine the behavior of a physical system over time. Additionally, understanding the behavior of sequences can help in making predictions and analyzing data.

4. Can a sequence both converge and diverge?

No, a sequence can either converge or diverge, but not both. A sequence cannot approach a specific limit while also growing infinitely. However, it is possible for different subsequences within a sequence to have different convergence or divergence behaviors.

5. Is it necessary for a sequence to converge?

No, it is not necessary for a sequence to converge. Some sequences may diverge, but this does not make them any less important or useful in mathematics and science. In fact, many important mathematical concepts, such as the Fibonacci sequence, are examples of divergent sequences.

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