Of course, using a summation technique to give the sum of a divergent series a meaning is the same sort of thing Euler was scoffed at when he employed geometric sums.
Say,
S = 1 + r + r^2 + r^3 + r^4 + .... + r^(n-1)
(we could've made the initial term something other than 1, say 'a', but this only requires multiplying by 'a' whenever S arises as is)
Then,
S - Sr = S (1 - r) = 1 + r + r^2 + .... + r^(n-1) - r - r^2 - .... - r^n = 1 - r^n
(since all the terms in between 'telescope' out)
Thus, for r not equal to 1,
S = (1 - r^n) / (1 - r)
Now for the infinite sum of S we need to find the limit of the sequence of partial sums,
lim_{n->infinity} S
Noting how r^n will only go to a finite value (zero to be precise) as n gets large without bound only when -1
lim_{n->infinity} S = 1/ (1 - r)
as the infinite sum in the traditional sense!
Euler, in playing around with this identity sought to remove the restriction of -1
1 + r + r^2 + .... + r^n + .... ad inf. := 1 / (1 - r) ---------- (1)
is valid for all values of r save for r = 1
This leads to some intriguing examples, for instance, r = 2 gives,
1 + 2 + 4 + 8 + ..... ad inf. = -1
and similar seemingly irreconcilable identities for other values of r.
However, as Hardy fittingly remarks in his seminal work on this subject 'Divergent Series', it is a mistake to think of Euler as a 'loose mathematician'. He acknowledged that what he was doing was no longer the standard summation procedure that was in use (this was before Cauchy made it all rigorous) and treated these identities almost as they would be treated in the modern theory of divergent series. If anything, he had the proper sort of reservations about the matter, as his dictum 'summa cujusque seriei est valor expressionis illius finitae,...' leads us to believe.
His belief that a divergent series must always be associated the same sum by different expressions kept him at bay from the rigorous theory, though his ideas were well in advance of their times and his reasons were different from many of his more orthodox contemporaries.
Euler thought in terms of limits of power series in his consideration of these series. Essentially, where the power series represented the infinite series, when the corresponding function attained a limit one would be able to associate that limit as the sum of the series.
Consider for instance,
1 + x + x^2 + .... + x^n + .... ad inf. := 1 / (1 - x)
valid for all x under say, the A sense of summation.
The finite values that appear in the right hand side even when the left hand side expression diverges in the traditional sense can be made sense of by considering it as the remainder from a process of algebraic long division (note that this is the same sort of singularity we removed in finding the Ramanujan sum of H).
This sort of consideration, while it can be made rigorous, is not without counter-intuitive backlashes. For instance, this implies that,
1 - 1 + 1 - 1 +....
and
1 + 0 - 1 + 1 + ....
while having essentially the same terms, yield different sums since the power series have different 'gaps'!
But then, considering the rearrangement theorem of Riemann we encountered early on for the Cauchy sense of convergence, this doesn't seem as exaggerated a bullet to bite after all.
Euler's justification would come much later with the advent of analytic continuation in the theory of functions of a complex variable. Where the domain of a given function can be extended under certain conditions.
If f and g are holomorphic on domains A and B respectively, and the functions coincide in the non-empty intersection of A and B, g is termed the analytic continuation of f to B and f is in turn termed the analytic continuation of g to A.
What is most striking here is that the analytic continuation whenever it exists is unique! It would seem Euler's attitude almost foresshadowed the implications of this powerful result.
What need be kept in mind is that while rigorous arguments in mathematics makes for the most refined of intellectual achievement, it should never bog us down and force us along a single linear track that we might find a naive comfort in.
In retrospect, Abel sounds disappointed as he writes them off in ushering in a new dawn of rigour-
“Divergent series are, in general, something terrible and it is a shame to base any proof on them.
We can prove anything by using them and they have caused so much misery and created so
many paradoxes. . . . . Finally my eyes were suddenly opened since, with the exception of the
simplest cases, for instance the geometric series, we hardly find, in mathematics, any infinite
series whose sum may be determined in a rigorous fashion, which means the most essential
part of mathematics has no foundation. For the most part, it is true that the results are
correct, which is very strange. I am working to find out why, a very interesting problem.”
Certainly, from this statement it is clear that Abel saw significant enough a problem to want to pursue it, though exactly the scarcity he remarks upon would become crucial for a more complete theory of functions as we have today (which was thanks to the efforts of Abel, Cauchy, et al.).
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