Abel's theorem explained

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]

Theorem

Let the Taylor series$$G\; (x)\; =\; \backslash sum\_^\backslash infty\; a\_k\; x^k$$be a power series with real coefficients

with radius of convergence Suppose that the series $$\backslash sum\_^\backslash infty\; a\_k$$ converges. Then is continuous from the left at that is,$$\backslash lim\_\; G(x)\; =\; \backslash sum\_^\backslash infty\; a\_k.$$

The same theorem holds for complex power series $$G(z)\; =\; \backslash sum\_^\backslash infty\; a\_k\; z^k,$$ provided that

entirely within a single Stolz sector, that is, a region of the open unit disk where$$|1-z|\; \backslash leq\; M(1-|z|)$$for some fixed finite . Without this restriction, the limit may fail to exist: for example, the power series$$\backslash sum\_\; \backslash frac\; n$$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

The Stolz sector

has explicit equation$$y^2\; =\; -\backslash frac$$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 .

Remarks

As an immediate consequence of this theorem, if

is any nonzero complex number for which the series $$\backslash sum\_^\backslash infty\; a\_k\; z^k$$ converges, then it follows that$$\backslash lim\_\; G(tz)\; =\; \backslash sum\_^\backslash infty\; a\_kz^k$$in which the limit is taken from below.

The theorem can also be generalized to account for sums which diverge to infinity. If$$\backslash sum\_^\backslash infty\; a\_k\; =\; \backslash infty$$then $$\backslash lim\_\; G(z)\; \backslash to\; \backslash infty.$$

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 $$\backslash frac.$$

At

the series is equal to but

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

: let $$G(x)\; =\; \backslash sum\_^\backslash infty\; a\_kx^k$$ be a power series with radius of convergence and suppose the series converges at Then is continuous from the left at that is,$$\backslash lim\_G(x)\; =\; G(R).$$

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,

) 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 $$a\_k\; =\; \backslash frac,$$we obtain by integrating the uniformly convergent geometric power series term by term on ; thus the series $$\backslash sum\_^\backslash infty\; \backslash frac$$ converges to by Abel's theorem. Similarly, $$\backslash sum\_^\backslash infty\; \backslash frac$$ 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$$G\_a(z)\; =\; (1-z)\backslash sum\_^\; s\_k\; z^k.$$

Given

pick large enough so that for all and note that$$\backslash left|(1-z)\backslash sum\_^\backslash infty\; s\_kz^k\; \backslash right|\; \backslash leq\; \backslash varepsilon\; |1-z|\backslash sum\_^\backslash infty\; |z|^k\; =\; \backslash varepsilon|1-z|\backslash frac$$

^n

< \varepsilon M when lies within the given Stolz angle. Whenever is sufficiently close to we haveso that when is both sufficiently close to 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.

Further reading

• Book: Ahlfors, Lars Valerian. Lars Ahlfors

. Lars Ahlfors. September 1, 1980. Complex Analysis. Third. McGraw Hill Higher Education. 41–42. 0-07-085008-9. - Ahlfors called it Abel's limit theorem.

External links

• (a more general look at Abelian theorems of this type)

Notes and References

1. Abel . Niels Henrik . 1826 . Untersuchungen über die Reihe

   1+

   	m
   	1

   x+

   	m ⋅ (m-1)
   	2 ⋅ 1

   x²+

   	m ⋅ (m-1) ⋅ (m-2)
   	3 ⋅ 2 ⋅ 1

   x³+\ldots

   u.s.w. . . 1 . 311–339 . Niels Henrik Abel.