Absolute convergence of series?

In summary, the conversation discusses how to show that if the series Ʃan from n=1 to ∞ converges absolutely, then the series Ʃan2 from n=1 to ∞ also converges absolutely. The idea is that if a_n is small, then a_n^2 is even smaller, making the squared series smaller than the original series. However, if a_n^2 is greater than 1, there is a problem. The original approach suggested in the conversation may not be helpful in solving this issue.
  • #1
SMA_01
218
0
The question is: Show that if [itex]\sum[/itex]an from n=1 to ∞ converges absolutely, then [itex]\sum[/itex]an2 from n=1 to converges absolutely.

I'm not sure which approach to take with this.

I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or positive. But I'm not sure where to go with this.

Any help is appreciated.

Thank you.
 
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  • #2
You know that if [itex]a_n[/itex] is small (=smaller than 1), then [itex]a_n^2[/itex] is even smaller. So the squared series is smaller than the original series.

Can you do something with this idea??
 
  • #3
So you mean a_n^2 would be bounded from above for smaller values? But what if it's greater than 1?
 
  • #4
SMA_01 said:
So you mean a_n^2 would be bounded from above for smaller values? But what if it's greater than 1?

Yes, if it's greater than 1 then there is a problem. Try to find a way to solve this.
 
  • #5
Do you think I can use what I stated in my first post?
 
  • #6
SMA_01 said:
Do you think I can use what I stated in my first post?

Uh, not really. Well,it's not wrong what you said, but it won't help you much further.
 

Related to Absolute convergence of series?

1. What is absolute convergence of series?

Absolute convergence of series is a mathematical term used to describe a series in which the sum of the absolute values of the terms is finite. In simpler terms, it means that the series converges even when the signs of the terms are ignored.

2. How is absolute convergence different from conditional convergence?

Conditional convergence refers to a series that converges, but only when the signs of the terms are taken into account. Absolute convergence, on the other hand, means the series converges regardless of the signs of the terms.

3. Why is absolute convergence important?

Absolute convergence is important because it guarantees that the series will converge to a finite value, regardless of the order in which the terms are added. This allows for simpler and more reliable calculations in mathematics and physics.

4. How do you test for absolute convergence?

To test for absolute convergence, one can use the Ratio Test or the Comparison Test. If the limit of the absolute value of the terms is less than 1, then the series is absolutely convergent. If the limit is greater than 1, the series is not absolutely convergent. If the limit is equal to 1, further tests may be needed.

5. Can a series be both absolutely and conditionally convergent?

Yes, a series can be both absolutely and conditionally convergent. This means that the series converges regardless of the signs of the terms, but also converges only when the signs are taken into account. An example of such a series is the alternating harmonic series.

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