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.
Abel's theorem
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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
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
1}">.
Without this restriction, the limit may fail to exist: for example, the power series
0}{frac {z^{3^{n}}-z^{2cdot 3^{n}}}{n}}}">
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
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.[citation needed] 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
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
0,"> 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[]
- Abel's summation formula – Integration by parts version of Abel's method for summation by parts
- Nachbin resummation
- Summation by parts – Theorem to simplify sums of products of sequences
Further reading[]
Ahlfors, Lars Valerian (September 1, 1980). Complex Analysis (Third ed.). McGraw Hill Higher Education. pp. 41–42. ISBN 0-07-085008-9. - Ahlfors called it Abel's limit theorem.
External links[]
- Abel summability at PlanetMath. (a more general look at Abelian theorems of this type)
- A.A. Zakharov (2001) [1994], "Abel summation method", Encyclopedia of Mathematics, EMS Press
- Weisstein, Eric W. "Abel's Convergence Theorem". MathWorld.
Si quieres conocer otros artículos parecidos a Abel's theorem puedes visitar la categoría Mathematical series.
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