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# Abel summability

Abel 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 $\sum_{{n=0}}^{\infty}a_{n}$, convergent or not, and use that series to define a power series

$f(r)=\sum_{{n=0}}^{\infty}a_{n}r^{n}.$ |

Note that for $|r|<1$ the
summability of $f(r)$ is easier to achieve than the summability of the
original series. Starting with this observation we say that the
series $\sum a_{n}$ is *Abel summable* if the defining series
for $f(r)$ is convergent for all $|r|<1$, and if $f(r)$ converges to
some limit $L$ as $r\rightarrow 1^{-}$. If this is so, we shall say
that $\sum a_{n}$ Abel converges to $L$.

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 $\sum_{{n=0}}^{\infty}a_{n}$ be a series; let

$s_{N}=a_{0}+\cdots+a_{N},\quad N\in\mathbb{N},$ |

denote the corresponding partial sums; and let $f(r)$ be the corresponding power series defined as above. If $\sum a_{n}$ is convergent, in the usual sense that the $s_{N}$ converge to some limit $L$ as $N\rightarrow\infty$, then the series is also Abel summable and $f(r)\rightarrow L$ as $r\rightarrow 1^{-}$.

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

$\sum_{{n=0}}^{\infty}(-1)^{n}.$ |

The corresponding power series is

$\frac{1}{1+r}=\sum_{{n=0}}^{\infty}(-1)^{n}r^{n}.$ |

Since

$\frac{1}{1+r}\rightarrow\frac{1}{2}\quad\text{as}\quad r\rightarrow 1^{-},$ |

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

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
illustrates^{1}^{1}We want the converse to be false; the whole idea
is to describe a method of summing certain divergent series!.
However, in the 1890’s
Tauber
proved the following partial converse.

###### Theorem 2 (Tauber).

Suppose that $\sum a_{n}$ is an Abel summable series and that $na_{n}\rightarrow 0$ as $n\rightarrow\infty$. Then, $\sum_{n}a_{n}$ 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.

## Mathematics Subject Classification

40G10*no label found*

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