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 seriesbe 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.
Note that the convergence of implies that the radius of convergence of the power series is at least 1, ensuring convergence for.
Also note that 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. 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 seriesconverges, 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
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument 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, whenwe 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:Let and Then substituting and performing a simple manipulation of the series 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.
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:
Note that in the absence of additional assumptions, the series might not converge when, hence the use of the limit inferior.