Let be a power series with real coefficients with radius of convergence. Suppose that the series converges. Then is continuous from the left at, i.e. The same theorem holds for complex power series provided that within a Stolz sector, that is, a region of the open unit disk where for some. 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 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. 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, i.e.
Applications
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument approaches 1 from below, even in cases where the radius of convergence,, of the power series is equal to 1 and we cannot be sure whether the limit should be finite or not. See e.g. 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.
After subtracting a constant from, we may assume that. Let. 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 1 we have so that when is both sufficiently close to 1 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.
