Divergent Series - Properties of Summation Methods

Properties of Summation Methods

Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an average of larger and larger initial terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. So in evaluating a = a₀ + a₁ + a₂ + ..., we work with the sequence s, where s₀ = a₀ and s_n+1 = s_n + a_n+1. In the convergent case, the sequence s approaches the limit a. A summation method can be seen as a function from a set of sequences of partial sums to values. If A is any summation method assigning values to a set of sequences, we may mechanically translate this to a series-summation method AΣ that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.

1. Regularity. A summation method is regular if, whenever the sequence s converges to x, A(s) = x. Equivalently, the corresponding series-summation method evaluates AΣ(a) = x.
2. Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that A(r + s) = A(r) + A(s) and A(ks) = k A(s), for k a scalar (real or complex.) Since the terms a_n = s_n+1 − s_n of the series a are linear functionals on the sequence s and vice versa, this is equivalent to AΣ being a linear functional on the terms of the series.
3. Stability. If s is a sequence starting from s₀ and s′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that s′_n = s_n+1 − s₀, then A(s) is defined if and only if A(s′) is defined, and A(s) = s₀ + A(s′). Equivalently, whenever a′_n = a_n+1 for all n, then AΣ(a) = a₀ + AΣ(a′).

The third condition is less important, and some significant methods, such as Borel summation, do not possess it.

One can also give a weaker alternative to the last condition.

1. Finite Re-indexability. If s and s′ are two sequences such that there exists a bijection such that s_i = s′_f(i) for all i, and if there exists some such that s_i = s′_i for all i > N, then A(s) = A(s′). (In other words, s′ is the same sequence as s, with only finitely many terms re-indexed.) Note that this is a weaker condition than Stability, because any summation method that exhibits Stability also exhibits Finite Re-indexability, but the converse is not true.

A desirable property for two distinct summation methods A and B to share is consistency: A and B are consistent if for every sequence s to which both assign a value, A(s) = B(s). If two methods are consistent, and one sums more series than the other, the one summing more series is stronger.

There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levin-type sequence transformations and Padé approximants, as well as the order-dependent mappings of perturbative series based on renormalization techniques.