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

272 Chapter 13. ABEL and OLIVIER on convergence tests
Since the series (13.2) was divergent, the series (13.3) would have to be divergent as
well (by the procedure above). On the other hand, the generalized criterion, when
applied to (13.3) gave
φ (n) an =
1
∑ n−1
k=1
,
1
φ(k)
which by the very divergence of (13.2) converged to zero for n → ∞. Thus, the se-
ries (13.3) produced a general counter example to ABEL’S generalization of OLIVIER’S
proposed convergence criterion.
By this very elegant proof, ABEL turned OLIVIER’S proposed criterion against itself
and it imploded. Thus, ABEL proved that no simple test of convergence of series could
be devised. Interpreted as a question of delineation of concepts, ABEL’S result thus
meant that the extent of the concept of convergent series was not easily determined by
external criteria.
13.4 More characterizations and tests of convergence
In its published form, ABEL’S answer to OLIVIER’S paper was a negative one, in the
sense that it refused a proposed theorem. However, in his notebooks, ABEL elabo-
rated some of the ideas found therein to such a degree as to produce new, positive
knowledge in the form of new characterizations and tests of convergence. 17
In his notebook draft, ABEL obtained his own version of a limit comparison theorem
which provided a necessary criterion for convergence. He claimed that if ∑ φ (n)
was a divergent series, and ∑ an was a convergent one, it would be necessary that
“the smallest among the limits of
an
φ(n) be zero.”18 ABEL’S proof was indirect: Under
the contrary assumption, he wrote un = pnφ (n) where pn ≥ α. Then
∑ un > ∑ αφ (n) = α ∑ φ (n) → ∞.
From this, ABEL obtained the second part of OLIVIER’S theorem which he had not
objected to: Because ∑ 1 n was known to be divergent, if ∑ an was to be convergent, it
would be necessary that nan vanished as n became infinite.
Pairs of convergent and divergent series. Also in the notebook, we find a general-
ization of the lemma 2 to the effect that the divergence of ∑ an implied the divergence
of the series ∑ an
s α n
where 0 ≤ α ≤ 1 (lemma 2 results from setting α = 1). A converse to
this result was also obtained when ABEL proved that if the series ∑ an was divergent,
then the series
17 (N. H. Abel, [1827] 1881).
18 (N. H. Abel, 1881, II, 198).
∞
an
∑
n=1 s 1+α
n