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

266 Chapter 13. ABEL and OLIVIER on convergence tests
OLIVIER’S curious inserted remark — that the n th group of terms with the same sign
could be considered instead of an — probably derived from the fact that within such a
group, reordering of the terms could not effect the convergence or sum of the series.
However, both in OLIVIER’S paper and in the present analysis, the important case
arises by considering individual terms.
For the historian, OLIVIER’S theorem requires an interpretation which is by no
means easy or unambiguous; for instance, what does it mean that “this circumstance
is a sign”? The main question in interpreting the theorem lies in this phrase, since
it could be read to mean that if nan → 0 then the series will always be convergent.
This was certainly the way it was interpreted by some of OLIVIER’S readers; however,
the phrasing is sufficiently weak to call for further investigation. In the following,
OLIVIER’S argument is outlined in order to illustrate how mathematicians in the early
19 th century still argued about infinite series. Then, to supplement the theorem and
its proof, a consideration of the examples to which it was applied is necessary before
a weighed interpretation of the theorem can be given.
13.1.1 OLIVIER’s first proof
OLIVIER gave two arguments which illustrate how he came to believe in his theorem.
The first argument was given immediately before the theorem was stated, whereas
the second one was prompted by ABEL’S objection to the theorem and printed as a
response to ABEL’S note. 6
In 1827, OLIVIER divided infinite series into three categories: convergent, indeter-
minate, and divergent. His definitions and the ensuing proofs are difficult to represent
fairly, because his concepts are different and vague and his style of reasoning is rather
verbal and leaves few hints on the unclear points. OLIVIER’S definition of convergent
series consisted of two requirements:
“One calls a series convergent which has the following two properties, namely:
that one finds its numerical value ever more exactly when one calculates successively
more terms and that by continuing the calculation indefinitely, one can approach
the true value of the entire series to any degree one wishes.”” 7
In OLIVIER’S definition, we see a curious and obscure mixture of the old and the
new concepts of convergence. At the same time, OLIVIER speaks of numerical approx-
5 “Donc si l’on trouve, que dans une série infinie, le produit du n me terme, ou du n me des groupes
de termes qui conservent le même signe, par n, est zéro, pour n = ∞, on peut regarder cette seule
circonstance comme une marque, que la série est convergente; et réciproquement, la série ne peut
pas être convergente, si le produit n.an n’est pas nul pour n = ∞.” (Olivier, 1827, 34).
6 (Olivier, 1828)
7 “On appelle convergente une série, qui a les deux propriétés suivantes, savoir: qu’on trouve sa valeur
numérique d’autant plus exactement, qu’on calcule successivement plusieurs termes, et qu’en
continuant indéfiniment ce calcul, on peut se rapprocher de la vraie valeur de la série totale à tel
degré qu’on voudra.” (Olivier, 1827, 31).