Absolute convergence

In summary: Here is a link to a series that diverges but does not converge absolutely:Here is a link to a series that diverges but does not converge absolutely: http://en.wikipedia.org/wiki/Alternating_series
  • #1
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If [tex]\sum x_n[/tex] converges absolutely, and the sequence (yn) is bounded, then the sum [tex]\sum x_n y_n[/tex] converges.

Find a counterexample that shows this isn't true when [tex]\sum x_n[/tex] is conditionally convergent.

I'm honestly not to sure where to begin with this one. I was thinking Monotone Convergence Theorem, but that might not be necessarily true for xn Any suggestions would be fantastic!
 
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  • #2
Here's one way. Take some concrete example of alternating series x_n, that converges, but not absolutely. Then start choosing values for y_n one by one, so that |y_n| < 1, and trying to make the value of the series x_n*y_n as large as possible. You should be able to get infinity out of it.
 
  • #3
jostpuur said:
Here's one way. Take some concrete example of alternating series x_n, that converges, but not absolutely. Then start choosing values for y_n one by one, so that |y_n| < 1, and trying to make the value of the series x_n*y_n as large as possible. You should be able to get infinity out of it.

I'm not entirely sure I follow you...
Are you suggesting I choose lim(sup{xn}) or lim(inf{xn}, which are both monotone decreasing and increasing respectively?
 
  • #4
Wow, ok nevermind:

We know [tex]\sum x_n[/tex] converges.
|yn| <= M, For any n contained in the naturals - Def of bounded.

Then,
[tex]\sum x_n y_n = \sum \left|x_n\right| \left|y_n\right| \leq M\sum \left|x_n\right|[/tex]

Therefore, the sum converges!

A counter example would be to let xn = [tex]\frac{(-1)^n}{n}[/tex] and yn = [tex](-1)^n[/tex]
 
  • #5
There is a really easy way to do this problem. What is one series you know that diverges, the one I am thinking of is probably one of the first examples you received as a divergent series when you first learned about series. Now is there a series that looks very similar to this one that converges, but not absolutely (perhaps an alternating series...). Then how can you turn this alternating series into the one which does not converge by choosing appropriate, and bounded yn?

Edit: Nevermind you figured it out.
 

What is absolute convergence?

Absolute convergence refers to a mathematical concept where the sum of a series of numbers or terms always converges to a definite value, regardless of the order in which the terms are arranged.

How is absolute convergence different from conditional convergence?

Conditional convergence is a mathematical concept where the sum of a series of numbers or terms only converges to a definite value if the terms are arranged in a specific order. Absolute convergence, on the other hand, always converges to a definite value regardless of the order of the terms.

What is the significance of absolute convergence in mathematics?

Absolute convergence is important in mathematics because it allows for the manipulation and rearrangement of series without changing the overall sum. This makes it a useful tool in various mathematical and scientific applications.

How can absolute convergence be tested for?

A series can be tested for absolute convergence by using various convergence tests, such as the ratio test, root test, or comparison tests. If a series passes one of these tests, it can be determined to be absolutely convergent.

Are there any real-life applications of absolute convergence?

Absolute convergence has many real-life applications, such as in economics, physics, and engineering. It is used to analyze and predict the behavior of systems and make accurate calculations in various fields.

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