http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Abel.htmlAbel summability is a generalized convergence criterion for power series. It extends the usual definition of the sum of a series, and gives a way of summing up certain divergent series. Let us start with a series , convergent or not, and use that series to define a power series
Note that for the summability of is easier to achieve than the summability of the original series. Starting with this observation we say that the series is Abel summable if the defining series for is convergent for all , and if converges to some limit as . If this is so, we shall say that Abel converges to .
Of course it is important to ask whether an ordinary convergent series is also Abel summable, and whether it converges to the same limit? This is true, and the result is known as Abel’s limit theorem, or simply as Abel’s theorem.
Theorem 1 (Abel)
Let be a series; let
denote the corresponding partial sums; and let be the corresponding power series defined as above. If is convergent, in the usual sense that the converge to some limit as , then the series is also Abel summable and as .
The standard example of a divergent series that is nonetheless Abel summable is the alternating series
The corresponding power series is
this otherwise divergent series Abel converges to .
Abel’s theorem is the prototype for a number of other theorems about convergence, which are collectively known in analysis as Abelian theorems. An important class of associated results are the so-called Tauberian theorems. These describe various convergence criteria, and sometimes provide partial converses for the various Abelian theorems.
The general converse to Abel’s theorem is false, as the example above illustrates11We want the converse to be false; the whole idea is to describe a method of summing certain divergent series!. However, in the 1890’s http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Tauber.htmlTauber proved the following partial converse.
Theorem 2 (Tauber)
Suppose that is an Abel summable series and that as . Then, is convergent in the ordinary sense as well.
The proof of the above theorem is not hard, but the same cannot be said of the more general Tauberian theorems. The more famous of these are due to Hardy, Hardy-Littlewood, Weiner, and Ikehara. In all cases, the conclusion is that a certain series or a certain integral is convergent. However, the proofs are lengthy and require sophisticated techniques. Ikehara’s theorem is especially noteworthy because it is used to prove the prime number theorem.
|Date of creation||2013-03-22 13:07:03|
|Last modified on||2013-03-22 13:07:03|
|Last modified by||rmilson (146)|