Proving Absolute Value Convergence of Sequence to A

In summary, the conversation discusses the convergence of a sequence {an} to a limit A, and whether the convergence of the absolute value of {an} to the absolute value of A implies the convergence of {an} to A. The conversation goes on to give an example of a sequence where the absolute value converges to a different value than the original sequence, showing that the convergence of the absolute value does not necessarily imply the convergence of the original sequence. The conversation also mentions a signature file, which can be edited by clicking on the "My PF" tab at the top of the page.
  • #1
IntroAnalysis
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0

Homework Statement


If the absolute value of a sequence, an converges to absolute value of A, does sequence, an necessarily converge to A?


Homework Equations


convergence: a sequence { an}n=1-->infinity, converges to A є R (A is called the limit of the sequence) iff for all є > 0, there exists an N є Natural, for all n[itex]\geq[/itex] N (│an - A│< є ).

Also, know that │ │a│ - │b││ [itex]\leq[/itex] │a - b│

The Attempt at a Solution


I've been trying to find a counterexample, but so far I haven't been able to. Any suggestions on this proof?
 
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  • #2
Consider a sequence whose sign changes frequently.
 
  • #3
So if my sequence = {(-1)n([itex]\frac{1}{n}[/itex])= {-1, 1/2, -1/3, 1/4, -1/5, ...} This is converging to 0.

If you have the absolute value of an, it also converges to 0. Remove the absolute value and you get the same convergence point. Maybe I don't have the sequence you had in mind?
 
  • #4
Try making the absolute value sequence converge to something other than 0.
 
  • #5
Try the sequence {1,-1,1,-1,1,-1,...}.

RGV
 
  • #6
Thanks, I finally get it. l (-1) l ^2 will converge to l 1l whereas without the absolute value this sequence never converges but bounces back and forth from -1 to 1.

Also, what is a signature file so that I can add some needed terms?
 
  • #7
IntroAnalysis said:
Also, what is a signature file so that I can add some needed terms?

Click on the "My PF" tab at the top of the page and you will see where to edit your signature file.
 

FAQ: Proving Absolute Value Convergence of Sequence to A

What is "Proving Absolute Value Convergence of Sequence to A"?

"Proving Absolute Value Convergence of Sequence to A" refers to the mathematical process of determining whether a given sequence of numbers approaches a specific value, known as "A", as the number of terms in the sequence increases. It involves analyzing the absolute values of the terms in the sequence to determine if they converge towards the value of A.

Why is it important to prove absolute value convergence of a sequence?

Proving absolute value convergence of a sequence is important because it allows us to determine the behavior of the sequence and whether or not it has a limit. This information is crucial in many areas of mathematics, including calculus, differential equations, and statistics, as well as in practical applications such as engineering and physics.

What is the difference between absolute value convergence and regular convergence of a sequence?

The main difference between absolute value convergence and regular convergence of a sequence is the approach used to determine whether a limit exists. In regular convergence, we consider the magnitude and direction of the terms in the sequence, while in absolute value convergence, we only consider the magnitude of the terms. This allows us to analyze more complicated sequences and determine their convergence more easily.

What are some common techniques for proving absolute value convergence of a sequence?

There are several techniques for proving absolute value convergence of a sequence, including the squeeze theorem, the limit comparison test, and the ratio test. These techniques involve comparing the given sequence to a simpler sequence with known convergence properties, and using properties of limits and inequalities to determine the convergence of the original sequence.

Can a sequence have absolute value convergence but not regular convergence?

Yes, it is possible for a sequence to have absolute value convergence but not regular convergence. This can occur when the terms in the sequence alternate between positive and negative values, causing the sequence to oscillate around the value of A instead of approaching it. In this case, the sequence would have absolute value convergence but not regular convergence.

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