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.

be a power series with real coefficients

The same theorem holds for complex power series

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

Without this restriction, the limit may fail to exist: for example, the power series

but is unbounded near any point of the form

is continuous on the real closed interval

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

and is plotted on the right for various values.

The left end of the sector is

On the right end, it becomes a cone with angle

is any nonzero complex number for which the series

The theorem can also be generalized to account for sums which diverge to infinity.

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

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

be a power series with radius of convergence

and suppose the series converges at

The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is,

from below, even in cases where the radius of convergence,

of the power series is equal to

Abel's theorem allows us to evaluate many series in closed form.

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

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.

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

lies within the given Stolz angle.

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.