Explanation of Limit Superior/Inferior

  • #1

Main Question or Discussion Point

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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  • #2
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  • #3
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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
HallsofIvy
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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.
 

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