RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

210 Chapter 11. CAUCHY’s new foundation for analysis
limit zero. However, by introducing symbols, the process under which the variable
vanished was obscured and the order in which limit processes were conducted was
not explicit.
11.3 Divergent series have no sum
When CAUCHY extended the procedure of analyzing the requirements for numerical
equality involving series, he was led to a conclusion which he knew would be painful
for his contemporaries to accept.
“It is true that in order to always remain faithful to these principles, I see myself
forced to accept multiple propositions which may appear a bit harsh at first
sight. For instance, in the sixth chapter, I announce that a divergent series has no
sum.” 10
CAUCHY’S treatment of series began with series of positive real terms (section
VI.2), was then extended to series of general real terms (section VI.3), before he went
on to treat series with complex terms (chapter IX). In the sixth chapter, CAUCHY elab-
orated his definition of convergence and his attitude toward divergent series.
“Let
sn = u0 + u1 + u2 + · · · + un−1
be the sum of the first n terms where n designates any integer. If, for ever increasing
values of n, the sum sn approaches a certain limit s indefinitely, the series is
said to be convergent and the above mentioned limit is called the sum of the series.
In the contrary case, if the sum sn does not approach any fixed limit when n
increases indefinitely, the series is divergent and no longer has a sum.” 11
As the quotations demonstrate, CAUCHY sought to limit the concept of “sum of a
series” to apply only to convergent series. This position was radicalized by ABEL in
his correspondence as we will see in section 12.3: ABEL wanted an outright ban on
divergent series and saw them as the creation of the Devil. To CAUCHY, who was also
a very creative mathematician outside fundamental issues, divergent series remained
of interest in asymptotic mathematics; only in questions of foundational nature, they
were not attributed any sum.
10 “Il est vrai que, pour rester constamment fidèle à ces principes, je me suis vu forcé d’admettre plusieurs
propositions qui paraîtront peut-être un peu dures au premier abord. Par exemple, j’énonce
dans le chapitre VI, qu’un série divergente n’a pas de somme [. . . ]” (A.-L. Cauchy, 1821a, iv).
11 “Soit
sn = u0 + u1 + u2 + · · · + un−1
la somme des n premiers termes, n désignant un nombre entier quelconque. Si, pour des valeurs
de n toujours croissantes, la somme sn s’approche indéfiniment d’une certaine limite s; la série sera
dite convergente, et la limite en question s’appellera la somme de la série. Au contraire, si, tandis
que n croît indéfiniment, la somme sn ne s’approche d’aucune limite fixe, la série sera divergente, et
n’aura plus de somme.” (ibid., 123).