What is Tannery's Theorem? I can't seem to find a definition of what it is. And how do you use it?
Yes, it is hard to find anything on this subject, but
According to one source (http://www.wpr.aaugonline.net/gazette/ [Broken]), E.H.Neville discussed Tannery's theorem in The Mathematical Gazette, July 1930 Issue, p. 166.
Apparently it is a thereom concerning convergence of an infinite series.
See also a reference to Tannery's theorem (see §63, page 161, Advanced Calculus by G.A. Gibson, MacMillan 1954; also the February 2002, Volume 109, 196-200 AMM article by Josef Hofbauer) from http://www.numbertheory.org/papers.html
and there is a problem set from a math course at DePaul University - http://condor.depaul.edu/~rjohnson/foma/exercises.pdf [Broken] - which refers to Tannery's theorem. This refers to another source: Johnsonbaugh/Pfaffenberger: Foundations of Mathematical Analysis.
Just a note, Johnsonbaugh/Pfaffenberger book doesn't actually have any info on Tannery's theorem, it appears that exercise set is just supplementary problems though it does ask you to prove Tannery's theorem. It looks simple enough to prove, and gives some conditions that will allow you to interchange an infinite sum with a limit.
Separate names with a comma.