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.