Explanation of Limit Superior/Inferior

[Living Laser]

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 { A_{}n, A_{}n+1, A_{}n+2, ...}

The uniqueness of sup makes this definition seem meaningless to me. Obviously misinterpreting something dramatically here but hopefully someone can set me straight! Thanks.

 

[TylerH]

Did you check Wikipedia? http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior Seems to make sense to me.

Maybe the illustration will help:
[URL]http://upload.wikimedia.org/wikipedia/commons/c/c8/LimSup.svg[/URL]

 

[ForMyThunder]

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).

 

[HallsofIvy]

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.