Abel's theorem

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.
Theorem
Let the Taylor series
be a power series with real coefficients with radius of convergence 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 point of the form so the value at is not the limit as tends to 1 in the whole open disk.
Note that 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. Abel's theorem allows us to say more, namely that is continuous on
Stolz sector
The Stolz sector has explicit formulaand 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. 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
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is) approaches from below, even in cases where the radius of convergence, of the power series is equal to and we cannot be sure whether the limit should be finite or not. See for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when
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
After subtracting a constant from we may assume that Let Then substituting and performing a simple manipulation of the series (summation by parts) results in
Given pick large enough so that for all and note that
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.
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.
See also
Further reading
- Ahlfors called it Abel's limit theorem.
External links
(a more general look at Abelian theorems of this type)
Theorems in real analysis
Theorems in complex analysis
Mathematical series
Niels Henrik Abel
Summability methods