Explanation of Limit Superior/Inferior

In summary, The conversation discusses the concept of lim superior/inferior and the confusion surrounding its definition. The speakers are looking for a clear explanation or resource to understand it better. One speaker shares their understanding of limsup as the "greatest limit that a subsequence can have," while another suggests checking Wikipedia for more information.
  • #1
Living Laser
1
0
I'm in the middle of a Complex Analysis course, working on the proof for a theorem to calculate radii of convergence for power series; a formula that utilises lim sup. Unfortunately, either I've forgotten lim superior/inferior from 1st year Analysis or I was never taught it in the first place. I think I have a vague intuitive notion of what it is (the point at which other points in the series cluster near the upper and lower bound respectively?), but can't seem to understand/derive that from the formalism. To make things worse, every account I've read gives conflicting definitions, and I'm struggling to see the equivalence between them. I was wondering if anybody had a recommendation for a definitive treatment (the analysis textbooks I own miss it out, and the ones I picked out in the library today only give short expositions as part of end of chapter exercises), or perhaps could help me out with their own explanation.

I think what was particularly confusing was a definition which defined lim sup as (sorry, new to the forum and still working out how to do subscript properly) inf(sub n>1) sup { [tex]A_{}n[/tex], [tex]A_{}n+1[/tex], [tex]A_{}n+2[/tex], ...}

The uniqueness of sup makes this definition seem meaningless to me. Obviously misinterpreting something dramatically here but hopefully someone can set me straight! Thanks.
 
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  • #3
I usually think of the limsup as the "greatest limit that a subsequence can have" and the liminf likewise defined. For instance if you let E denote the set of all limits of convergent subsequences, then limsup is exactly max(E) and liminf is exactly min(E).
 
  • #4
Strictly speaking the limsup is the supremum (least upper bound) of the set of all subsequential limits. It is the "greatest limit that a subsequence can have" only if that least upper bound happens to be in the set of subsequential limits itself.
 

What is the limit superior/inferior?

The limit superior and limit inferior are concepts in mathematical analysis that describe the behavior of a sequence of numbers as it approaches infinity. The limit superior is the largest number that the sequence can approach, while the limit inferior is the smallest number. In other words, the limit superior is the supremum (least upper bound) of the sequence, and the limit inferior is the infimum (greatest lower bound).

How is the limit superior/inferior calculated?

The limit superior and limit inferior can be calculated by taking the supremum and infimum, respectively, of the set of all possible limits of subsequences of the original sequence. In other words, we look at all the numbers that the sequence can approach, and then take the largest (limit superior) or smallest (limit inferior) of those numbers.

What is the significance of the limit superior/inferior?

The limit superior and limit inferior are important concepts in mathematical analysis as they provide information about the behavior of a sequence as it approaches infinity. They can be used to determine whether a sequence converges or diverges, and to find the rate of convergence or divergence.

How are the limit superior/inferior and limit points related?

The limit superior and limit inferior are closely related to the concept of limit points. A limit point is a number that a sequence can approach infinitely many times. The limit superior is the largest limit point, while the limit inferior is the smallest limit point. In other words, the limit superior and limit inferior describe the behavior of a sequence at its limit points.

Can the limit superior/inferior be infinite?

Yes, the limit superior and limit inferior can be infinite. This occurs when the sequence has no upper or lower bound, respectively. For example, in the sequence {1, 2, 3, ...}, the limit superior is infinite as the sequence continues to increase without bound. Similarly, in the sequence {-1, -2, -3, ...}, the limit inferior is infinite as the sequence continues to decrease without bound.

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