The methods of Borel, Abel and Cesaro, Voronoi & Norlund part III

We now come to the last part of the present series regarding summability methods. The methods we shall presently consider is due to Norlund and are known as Norlund methods. The method of Cesaro means which we shall consider later is an elementary case of these. A nearly equivalent definition of these had already been given by Voronoi which is why we mention him here.

Say we have a sequence of weights, (w_k) where k runs from 0 to n, with only the restriction that each term is non-negative and, for our convenience, w_0 is never zero.

We consider the sequence of partial sums of the form s_n = a_0 + a_1 + a_2 + ... + a_n

and construct the sequence t_n such that,

t_n = [(w_n).(s_0) + (w_(n-1)).(s_1) + (w_(n-2)).(s_2) + ... + (w_0).(s_n)] / (w_0 + w_1 + w_2 + ... + w_n)

and consider the limiting value of this new sequence as the sum, i.e-

a_0 + a_1 + a_2 + ... = s , (N, w_k)

if,

lim_{n -> infinity} t_n = s

Of course, an immediate concern here is where regularity holds given the freedom we have with assigning the weights and thus the generality of this method. However, it may be shown that regularity holds given,

w_n / (w_0 + w_1 + w_2 + ... + w_n) -> 0

and that any two such regular Norlund methods are consistent with each other, i.e- summing a series to the same sum given both limits exist.

Indeed, for a regular Norlund method, an Abelian theorem holds which draws the connection between Abel's method and Norlund's.

i.e-

If (N, w_k) is regular and a_0 + a_1 + a_2 + ... + a_n = s , (N, w_k) then the series given by,

f(x) = a_0 + (a_1).x + (a_2).x^2 + ... + (a_n).x^n + ...

has a positive radius of convergence and f(x) is analytic and regular for x between 0 (inclusive) and 1.

lim_{x -> 1^-} f(x) = s

Which means that if f(x) does converge for x between 0 and 1 the (N, w_k) method gives the same sum as the A method.

For w_k = 1 for all k we have a special case we call Cesaro summation to (C, 1) and with,

w_k = (n + m - 1)_C_(m - 1) 

we have the more general (C, k) method of Cesaro. (note that the notation n_C_k stands for the binomial coefficient nCk)

The (C, 1) method for instance, sums the series,

1 - 1 + 1 -1 + ... to 1/2 by averaging the partial sums when taking the 'Cesaro limit'.

A Diverging Appeal...

From abhorred 'inventions of the devil' to rigorously defined entities, we have briefly glimpsed the passage of divergent series through history. What the episode highlights is the freedom that comes of definitions that need not abide by intuitive notions on how a mathematical entity ought to be.

Restrictions however, are not to be entirely disposed of whenever it is sought to pursue interesting results. A method of summation is certainly appealing when it is general and not overly contrived as Hardy remarks in 'Divergent Series'.

There is also the matter of 'regularity'. Which is a first useful criteria by which to classify a given method. By claiming a method to be regular we mean that the sum is associates with a series will correspond to the sum in the traditional sense whenever the latter is not infinite. This serves to 'generalise' the notion of a sum.

We immediately note that the Ramanujan sense of sum we met previously is not a regular method. The finer points of that method are best seen in connection to the Euler MacLaurin summation formula which we shall come across later.

Are there any more discriminating properties by which we can classify our study of divergent series?

The following postulates are often useful:

1) First postulate of linearity

If a_0 + a_1 + a_2 + ... = S, then
k.a_0 + k.a_1 + k.a_2 + .... = kS

2) Second postulate of linearity

If a_0 + a_1 + a_2 + ... = S and b_0 + b_1 + b_2 + ... = T, then
(a_0 + b_0) + (a_1 + b_1) + ... = S + T

3) Postulate of stability

If a_0 + a_1 + a_2 + ... = S then a_1 + a_2 + ... = S - a_0 and vice versa

They are embodied in a large number of the methods we employ.

We shall conclude this discussion for now upon considering three interesting senses of sum.