Abel summability

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Abel.htmlAbelsummability is a generalized convergence criterion for power series.It extends the usual definition of the sum of a series, and gives away of summing up certain divergent series. Let us start with aseries ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$, convergent or not, and use that seriesto define a power series

$$f(r)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{r}^n.$$

Note that for thesummability of $f(r)$ is easier to achieve than the summability of theoriginal series. Starting with this observation we say that theseries $\sum a_n$ is Abel summable if the defining seriesfor $f(r)$ is convergent for all , and if $f(r)$ converges tosome limit $L$ as $r\to 1^{-}$. If this is so, we shall saythat $\sum a_n$ Abel converges to $L$.

Of course it is important to ask whether an ordinary convergent seriesis 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 ${\mathrm{\sum }}_{n\mathrm{=}\mathrm{0}}^{\mathrm{\infty }}{a}_n$ be a series; let

$$s_N=a_0+\mathrm{\cdots }+a_N,N\in \mathbb{N},$$

denote the correspondingpartial sums; and let $f\mathit{}\mathrm{(}r\mathrm{)}$ be the corresponding power seriesdefined as above. If $\mathrm{\sum }{a}_n$ is convergent, in theusual sense that the $s_N$ converge to some limit $L$ as$N\mathrm{\to }\mathrm{\infty }$, then the series is also Abel summable and$f\mathit{}\mathrm{(}r\mathrm{)}\mathrm{\to }L$ as $r\mathrm{\to }{\mathrm{1}}^{\mathrm{-}}$.

The standard example of a divergent series that is nonetheless Abelsummable is the alternating series

$$\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n.$$

The corresponding power series is

$$\frac{1}{1+r}=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n{r}^n.$$

Since

$$\frac{1}{1+r}\to \frac{1}{2}\mathit{\hspace{1em}}\text{as}\mathit{\hspace{1em}}r\to 1^{-},$$

this otherwise divergent series Abel converges to $\frac{1}{2}$.

Abel’s theorem is the prototype for a number of other theorems aboutconvergence, which are collectively known in analysis as Abeliantheorems. An important class of associated results are the so-calledTauberian theorems. These describe various convergence criteria, andsometimes provide partial converses for the various Abelian theorems.

The general converse to Abel’s theorem is false, as the example aboveillustrates.However, in the 1890’shttp://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Tauber.htmlTauberproved the following partial converse.

Theorem 2 (Tauber)

Suppose that $\mathrm{\sum }{a}_n$ is an Abel summable series and that $n\mathit{}{a}_n\mathrm{\to }\mathrm{0}$ as $n\mathrm{\to }\mathrm{\infty }$. Then, ${\mathrm{\sum }}_n{a}_n$ isconvergent in the ordinary sense as well.

The proof of the above theorem is not hard, but the same cannot besaid of the more general Tauberian theorems. The more famous of theseare due to Hardy, Hardy-Littlewood, Weiner, and Ikehara. In allcases, the conclusion is that a certain series or a certain integralis convergent. However, the proofs are lengthy and requiresophisticated techniques. Ikehara’s theorem is especially noteworthybecause it is used to prove the prime number theorem.