Summation of Divergent Series and Integration of Divergent Integrals

The attribution of some generalized sum to an infinite series that lacks an ordinary sum is known as the summation of a divergent series. The integration of a divergent integral, which lacks a definite value, involves the attribution of some value to the integral. Divergent series may be obtained when conditionally convergent series are multiplied, when functions are expanded in Fourier series, when series of functions are differentiated or integrated, and so on. Divergent series and integrals are often encountered in various problems in modern physics—for example, in the theory of electromagnetic fields.

In many cases, a sum can be assigned to a divergent series or a value can be associated with a divergent integral. In other words, a sum or value, in some generalized sense, can be found that has some of the basic properties of the ordinary sum of a convergent series or the ordinary value of a convergent integral. Usually, if the sum of the series

is S and the sum of the series

is T, it is required that the sum of the series

be λS μT and that the sum of the series

be S – a₀. Moreover, regular methods of summation are most often considered—that is, methods that yield for every convergent series its ordinary sum.

In most summation methods, a divergent series is regarded as the limit, in some sense, of a convergent series. More precisely, every term of the series

is multiplied by some factor λ_n(t) such that a convergent series

with sum δ(t) is then obtained. The factors λ_n(t) are chosen so that, for every fixed n, the limit of λ_n(t) is 1 for some continuous or discrete variation of the parameter t. The terms of series (2) will then tend to the corresponding terms of series (1). If δ(t) has a limit, this limit is called the generalized sum, of the given series, corresponding to the given choice of the factor or to the given method of summation. Suppose, for example, λ_n(t) = 1 for n ≤ t and λ_n(t) = 0 for n > t. If t → ∞ the ordinary concept of the sum of a series is obtained. The Abel summation method is obtained when λ_n(t) = tⁿ, for t < 1, and t → 1.

Some summation methods are concerned not with the result of the multiplication of the terms of a series by λ_n(t) but with the corresponding changes in the partial sums of the series. For example, in the method of arithmetic means, which is also known as the Cesàro method of order 1,

where

This method corresponds to the choice λ_m = (m – n + 1)/(m + 1) for n ≤ m and λ_n(m) = 0 for n > m. Suppose and ; furthermore, let and

exists, the series is said to be summable to A by the Cesàro method of order k. Cesàro methods of fractional order are also considered. The power of the Cesàro method increases with increasing k—that is, the set of series summable by the method expands as k increases.

Every series that is summable by a Cesàro method of some order is also summable by the Abel method; moreover, the sum is the same. For example, the series 1 – 1 + 1– ... + (–1)^n–1+ ... is summable to ½ by the Abel method, since

and

The Cesàro method yields the same value, since s_2n = 1, s_2n+1 = 0, σ_2n = (n + 1)/(2n + 1), σ_2n+1 = ½, and

The Cesàro and Abel methods are used in the theory of trigonometric series to determine functions in terms of their Fourier expansions, since the Fourier series of any continuous function is summable to the function by the Cesàro method of order 1 and, consequently, by the Abel method.

In 1901, G. F. Voronoi proposed a summation method that includes all the Cesàro methods as special cases. Suppose p_n ≥ 0, p₀ = 0, and P_n = p₀ + p₁ + ... + p_n. The limit

is called the generalized Voronoi sum of the series. The Voronoi method is regular if

In 1911, the German mathematician O. Toeplitz found necessary and sufficient conditions that must be satisfied by the triangular matrix ║a_mn║, where a_mn = 0 when n > m, in order for the summation method defined by the formula

to be regular. The Polish mathematician H. Steinhaus generalized these conditions to the case of square matrices.

An important role is played in the theory of analytic functions by the Borel summation method, which permits a function defined by a power series to be analytically continued outside the circle of convergence. S. N. Bernshtein and the German mathematician W. Rogosinski proposed an important summation method for trigonometric series. Bernshtein used the method to obtain convergent interpolation processes.

The theory of the integration of divergent integrals is similar to the theory of the summation of divergent series. For example, if the integral

diverges and

exists, the first integral is said to be summable to A by the Cesàro method of order λ.