I've been browsing in a fantastic book by V. S. Varadarajan, Euler Through Time: A New Look at Old Themes.
One of the things that mathematicians do occasionally is put down the old guys, even people like Euler, for stupid reasons. Varadarajan's book—which doesn't do this at all—gave me some ammunition I'd been curious to have myself concerning the viewpoint of 18th century mathematicians on divergent series.
Here's a question: what is the sum of the infinite series
1-1+1-1+1-1+ ... ?
It's customary for 20th (and 21st) century people to say "Tsk—tsk—first you have to tell me what the sum means, then I'll (try to) tell you what it is". It's customary to say that the old guys like the Bernoullis and Euler "lacked rigor" because they took these questions at face value, and tried to find their natural meaning.
Maybe.
Here's a fascinating extract from a 1760 paper by Euler, reproduced in Varadarajan's book:
Notable enough, however, are the controversies over the series 1-1+1-1 ... whose sum was given by Leibniz as 1/2, although others disagree. No one has yet assigned another value to that sum, and so the controversy turns on the question whether the series of this type have a certain sum. Understanding of this question is to be sought in the word "sum"; this idea, if thus conceived—namely the sum of a series said to be that quantity to which it is brought closer as more terms of the series are taken—has relevance only for convergent series, and we should in general give up this idea of sum for divergent series. Wherefore, those who thus define a sum cannot be blamed if they claim they are unable to assign a sum to a series. On the other hand, as series in analysis arise from the expansion of fractions or irrational quantities or even transcendentals, it will in turn be permissible in calculation to substitute in place of such a series that quantity out of which it is produced...
OK—so this isn't a statement of the Tauberian results, but still, it reads like Euler has his heart in exactly the right place. How could it be otherwise? Varadarajan earlier points out this quotation from G. H. Hardy (a 20th century mathematician who was an expert on divergent series, among other things...)
It does not occur to a modern mathematician that a collection of mathematical symbols should have a 'meaning' until one has been assigned by definition. It was not a triviality even to the greatest mathematicians of the 18th century. They had not the habit of definition: it was not natural to them to say 'by X we mean Y' .... It is broadly true to say that mathematicians before Cauchy asked not 'How shall we define 1-1+1-1+1- ...' but 'What is 1-1+1-1+...?'
Varadarajan comments on Hardy (pg 128):
In many ways the great exception to this was Euler. He took divergent series seriously, used them to obtain beautiful results, and had definite views aobut what is meant by the sum of a divergent series and how they should be used...
Right on!
