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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 a = {ak: k ≥ 0} be any sequence of real or complex numbers and let

$$G_a(z) = \sum_{k=0}^{\infty} a_k z^k\!$$

be the power series with coefficients a. Suppose that the series $$\sum_{k=0}^\infty a_k\!$$ converges. Then

$$\lim_{z\rightarrow 1^-} G_a(z) = \sum_{k=0}^{\infty} a_k,\qquad (*)\!$$

where the variable z is supposed to be real, or, more generally, to lie within any Stoltz angle, that is, a region of the open unit disk where

$$|1-z|\leq M(1-|z|) \,$$

for some M. Without this restriction, the limit may fail to exist.

Note that $$G_a(z)$$ is continuous on the real closed interval [0, t] for t < 1, 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 $$G_a(z)$$ is continuous on [0, 1].
Remark

As an immediate consequence of this theorem, if z is any nonzero complex number for which the series $$\sum_{k=0}^\infty a_k z^k\!$$ converges, then it follows that

$$\lim_{t\to 1^{-}} G_a(tz) = \sum_{k=0}^{\infty} a_kz^k\!$$

in which the limit is taken from below.
Applications

The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (i.e. z) approaches 1 from below, even in cases where the radius of convergence, R, 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 a_k = (-1)^k/(k+1), we obtain $$G_a(z) = \ln(1+z)/z for 0 < z < 1$$ , by integrating the uniformly convergent geometric power series term by term on [-z, 0]; thus the series \sum_{k=0}^\infty (-1)^k/(k+1)\! converges to ln(2) by Abel's theorem. Similarly, \sum_{k=0}^\infty (-1)^k/(2k+1)\! converges to arctan(1) = \pi/4 .

$$G_a(z)$$ is called the generating function of the sequence a. 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 a_0 \!, we may assume that $$\sum_{k=0}^\infty a_k=0\!$$. Let $$s_n=\sum_{k=0}^n a_k\!.$$ Then substituting $$a_k=s_k-s_{k-1}\!$$ and performing a simple manipulation of the series results in

$$G_a(z) = (1-z)\sum_{k=0}^{\infty} s_k z^k.\!$$

Given $$\epsilon > 0\!$$, pick n large enough so that $$|s_k| < \epsilon\!$$ for all $$k\ge n\!$$ and note that

$$\left|(1-z)\sum_{k=n}^\infty s_kz^k \right| \le \epsilon |1-z|\sum_{k=n}^\infty |z|^k = \epsilon|1-z|\frac{|z|^n}{1-|z|} < \epsilon M \!$$

when z lies within the given Stoltz angle. Whenever z is sufficiently close to 1 we have

$$\left|(1-z)\sum_{k=0}^{n-1} s_kz^k \right| < \epsilon$$ ,

so that $$|G_a(z)| < (M+1)\epsilon \!$$ when z is both sufficiently close to 1 and within the Stoltz 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.

Summation by parts
Abel's summation formula
Nachbin resummation