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.

Theorem
Let the Taylor series $$G (x) = \sum_{k=0}^\infty a_k x^k$$ be a power series with real coefficients $$a_k$$ with radius of convergence $$1.$$ Suppose that the series $$\sum_{k=0}^\infty a_k$$ converges. Then $$G(x)$$ is continuous from the left at $$x = 1,$$ that is, $$\lim_{x\to 1^-} G(x) = \sum_{k=0}^\infty a_k.$$

The same theorem holds for complex power series $$G(z) = \sum_{k=0}^\infty a_k z^k,$$ provided that $$z \to 1$$ entirely within a single Stolz sector, that is, a region of the open unit disk where $$|1-z| \leq M(1-|z|)$$ for some fixed finite $$M > 1$$. Without this restriction, the limit may fail to exist: for example, the power series $$\sum_{n>0} \frac{z^{3^n}-z^{2\cdot 3^n}} n$$ converges to $$0$$ at $$z = 1,$$ but is unbounded near any point of the form $$e^{\pi i/3^n},$$ so the value at $$z = 1$$ is not the limit as $$z$$ tends to 1 in the whole open disk.

Note that $$G(z)$$ 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 the restriction of $$G(z)$$ to $$[0, 1]$$ is continuous.

Stolz sector
The Stolz sector $$|1-z|\leq M(1-|z|)$$ has explicit equation$$y^2 = -\frac{M^4 (x^2 - 1) - 2 M^2 ((x - 1) x + 1) + 2 \sqrt{M^4 (-2 M^2 (x - 1) + 2 x - 1)} + (x - 1)^2}{(M^2 - 1)^2}$$and is plotted on the right for various values.

The left end of the sector is $$x = \frac{1-M}{1+M}$$, and the right end is $$x=1$$. On the right end, it becomes a cone with angle $$2\theta$$ where $$\cos\theta = \frac{1}{M}$$.

Remarks
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 $$\lim_{t\to 1^{-}} G(tz) = \sum_{k=0}^\infty a_kz^k$$ in which the limit is taken from below.

The theorem can also be generalized to account for sums which diverge to infinity. If $$\sum_{k=0}^\infty a_k = \infty$$ then $$\lim_{z\to 1^{-}} G(z) \to \infty.$$

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 $$\frac{1}{1+z}.$$

At $$z = 1$$ the series is equal to $$1 - 1 + 1 - 1 + \cdots,$$ but $$\tfrac{1}{1+1} = \tfrac{1}{2}.$$

We also remark the theorem holds for radii of convergence other than $$R = 1$$: let $$G(x) = \sum_{k=0}^\infty a_kx^k$$ be a power series with radius of convergence $$R,$$ and suppose the series converges at $$x = R.$$ Then $$G(x)$$ is continuous from the left at $$x = R,$$ that is, $$\lim_{x\to R^-}G(x) = G(R).$$

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, $$z$$) 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 for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when $$a_k = \frac{(-1)^k}{k+1},$$ we obtain $$G_a(z) = \frac{\ln(1+z)}{z}, \qquad 0 < z < 1,$$ by integrating the uniformly convergent geometric power series term by term on $$[-z, 0]$$; thus the series $$\sum_{k=0}^\infty \frac{(-1)^k}{k+1}$$ converges to $$\ln 2$$ by Abel's theorem. Similarly, $$\sum_{k=0}^\infty \frac{(-1)^k}{2k+1}$$ converges to $$\arctan 1 = \tfrac{\pi}{4}.$$

$$G_a(z)$$ 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 (summation by parts) results in $$G_a(z) = (1-z)\sum_{k=0}^{\infty} s_k z^k.$$

Given $$\varepsilon > 0,$$ pick $$n$$ large enough so that $$|s_k| < \varepsilon$$ for all $$k \geq n$$ and note that $$\left|(1-z)\sum_{k=n}^\infty s_kz^k \right| \leq \varepsilon |1-z|\sum_{k=n}^\infty |z|^k = \varepsilon|1-z|\frac{|z|^n}{1-|z|} < \varepsilon M $$ when $$z$$ lies within the given Stolz angle. Whenever $$z$$ is sufficiently close to $$1$$ we have $$\left|(1-z)\sum_{k=0}^{n-1} s_kz^k \right| < \varepsilon,$$ so that $$\left|G_a(z)\right| < (M+1) \varepsilon$$ when $$z$$ 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.