Abel's theorem

In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel, who proved it in 1826.

Theorem

Let the Taylor series

be a power series with real coefficients . Suppose that the series

converges. Then is continuous from the left at that is,

The same theorem holds for complex power series

provided that entirely within a single Stolz sector, that is, a region of the open unit disk where

for some fixed finite . Without this restriction, the limit may fail to exist: for example, the power series

converges to at but is unbounded near any other point of the form so the value at is not the limit as tends to 1 in the whole open disk.

The convergence of implies that the radius of convergence of the power series is at least 1, ensuring convergence for .

Also, by the uniform limit theorem, is continuous on the real closed interval for by virtue of the uniform convergence of the series on compact subsets of the disk of convergence (by the Weierstrass M-test). Abel's theorem allows us to say more, namely that the restriction of to is continuous.

Stolz sector

The Stolz sector has explicit equationand is plotted on the right for various values.

The left end of the sector is , and the right end is . On the right end, it becomes a cone with angle where .

Remarks

As an immediate consequence of this theorem, if is any nonzero complex number for which the series

converges, then it follows that

in which the limit is taken from below.

The theorem can also be generalized to account for sums which diverge to infinity; see below. If

then

However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for

At the series is equal to but

We also remark the theorem holds for radii of convergence other than : let

be a power series with radius of convergence and suppose the series converges at Then is continuous from the left at that is,

Applications

Abel's theorem allows us to evaluate many series in closed form if we can first prove that they converge. For example, consider the sequence

The sum converges by the alternating series test, so we can apply Abel's theorem to the corresponding generating function . We obtain

by integrating the uniformly convergent geometric power series term by term on ; thus the series

converges to by Abel's theorem. Similarly,

converges to

is called the generating function of the sequence Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton–Watson processes.

Outline of proof Source:Ahlfors 1980, pp. 41-42

Let and Then substituting and performing a simple manipulation of the series (summation by parts) results in

Given pick large enough so that for all We have

when lies within the given Stolz angle. Whenever is sufficiently close to we have

so that when is both sufficiently close to and within the Stolz angle.

Divergent case To prove the case for , let and . Since diverges to , we can find, for any , a such that for all . We write, for :

Since the first term on the right hand side vanishes as , we can find an such that it exceeds whenever . The second term may be estimated by:

Hence, if we let , then for this exceeds . Combining, we get, for any and :

This establishes:

In the absence of additional assumptions, the series might not converge when , hence the use of the limit inferior.

Related concepts

Converses to a theorem like Abel's are called Tauberian theorems: There is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type.

Generalizations

In the real case, the functions may be replaced by bounded non-negative functions monotonically decreasing with (i.e., ) such that for all .Whittaker and Watson, 2021, p. 48 This follows from Abel's uniform convergence test and applying the uniform limit theorem.

Abel's theorem

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