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.[1]



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 the restriction of   to   is continuous.

Stolz sector

20 Stolz sectors, for   ranging from 1.01 to 10. The red lines are the tangents to the cone at the right end.

The Stolz sector   has explicit equation and 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  .



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,  



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.


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 – Theorem bounding the growth rate of analytic functions
  • 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.


  1. ^ Abel, Niels Henrik (1826). "Untersuchungen über die Reihe   u.s.w.". J. Reine Angew. Math. 1: 311–339.