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

13.1. OLIVIER’s theorem 267
imation in language similar to A.-L. CAUCHY’S (1789–1857) and of the “true value”
of the series which resembles the Eulerian formal equality between functions.
OLIVIER separated non-convergent series into indeterminate and divergent ones:
“On the contrary, one calls a series indeterminate if continuing the calculation
of terms does not make it approach anything.
And one calls a series divergent in which the successive terms, added together,
produces results which differ more and more from the true value of the series.” 8
This distinction between two types on non-convergent series was probably inspired by
the discussion of Poisson’s example in which OLIVIER also participated without making
any noticeable contributions. 9
OLIVIER proceeded to express the two criteria of his definition of convergence in a
slightly different form. First, he observed, the terms of the series (or groups of terms
with the same sign) had to constantly decrease. Second, the sum of terms after the n th
term, i.e. the tail of the series, had to be zero for n = ∞. He gave similar translations
of the concepts of indeterminate and divergent series.
To obtain his theorem, OLIVIER first investigated the first condition concerning the
vanishing of the terms. He stated that this condition would always be satisfied if the
ratio of consecutive terms was always less than one. Thus, he apparently missed out
on cases in which an+1 an < 1 but lim an+1 an
= 1. For the second condition to also be
fulfilled, OLIVIER noted that it would be necessary and sufficient that nan → 0 for
n → ∞.
Under hypothesis nan → 0 and the further assumption that the terms an vanish as
n increases, OLIVIER claimed that
if the series was written as
R ≤ nan
a1 + a2 + · · · + an + R.
And thus, the vanishing of the tail R followed. How OLIVIER came to this [false] belief
will be clearer below.
On the other hand, the tail R could not vanish without nan also vanishing, OLIVIER
claimed. Using the constantly decreasing nature of the terms, OLIVIER found
na2n ≤ R ≤ nan
where R suddenly meant “the sum of n terms which follows after the n th term.” 10
Consequently, if nan vanished, so did R and the convergence of the series was secured.
8 “Au contraire, on appelle indéterminée une série, qui ne donne aucun rapprochement, en continuant
le calcul des termes.
Et on appelle divergente une série, dont les termes suivants, ajoutés aux précédents, ne donnent
que des résultats, qui s’éloignent plus en plus de vraie valeur de la série.” (ibid., 31).
9 (Olivier, 1826b). For a contemporary evaluation, see ([Saigey], 1826, 112).
10 “[. . . ] R, ou la somme des n termes qui suivent le n me terme.” (Olivier, 1827, 34).