Abel summability

Abel summability

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Abel.html

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

∑

n

=

0

∞

a

n

,

convergent

or not, and use that series
to define a power series

f

⁢

(

r

)

=

∑

n

=

0

∞

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

∑

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

→

1

-

. If this is so, we shall say
that

∑

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

∑

n

=

0

∞

a

n

be a series; let

s

N

=

a

0

+

⋯

+

a

N

,

N

∈

ℕ

,

denote the corresponding

partial sums

; and let

f

⁢

(

r

)

be the corresponding power series
defined as above. If

∑

a

n

is convergent, in the
usual sense that the

s

N

converge to some limit

L

as

N

→

∞

, then the series is also Abel summable and

f

⁢

(

r

)

→

L

as

r

→

1

-

.

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

alternating series

∑

n

=

0

∞

(

-

1

)

n

.

The corresponding power series is

1

1

+

r

=

∑

n

=

0

∞

(

-

1

)

n

⁢

r

n

.

Since

1

1

+

r

→

1

2

as

r

→

1

-

,

this otherwise divergent series Abel converges to

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

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Tauber.html

Tauber
proved the following partial converse.

Theorem 2

(Tauber)

Suppose that

∑

a

n

is an Abel summable series and that

n

⁢

a

n

→

0

as

n

→

∞

. Then,

∑

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

.

Title

Abel summability

Canonical name

AbelSummability

Date of creation

2013-03-22 13:07:03

Last modified on

2013-03-22 13:07:03

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Definition

Classification

msc 40G10

Related topic

CesaroSummability

Related topic

AbelsLimitTheorem

Defines

Abelian theorem

Defines

Tauberian theorem

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