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

252 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
and whose sum is
will also be convergent.” 58
(v0 + v1 + v2 + . . . ) × � v ′ 0 + v ′ 1 + v ′ 2 + . . . �
As it appears, the theorem consists of two halves each contributing a distinct con-
clusion:
1. Convergence of terms implies convergence of numerical terms.
2. Convergence of the Cauchy product toward the correct sum.
The first of these two conclusions is wrong, and it is difficult to explain the mishap
in ABEL’S presentation. In the collected works, it has been corrected without com-
ments by replacing “so sind die Reihen [. . . ] ebenfalls noch convergent” with “si les
séries [. . . ] sont de même convergente”. 59 In the proof, ABEL does not give argu-
ments for the first part of the supposed theorem; instead it is used as an assumption
in proving the Cauchy product theorem.
Thus, based on ABEL’S proof, SYLOW and LIE attributed the mishap to a slip of the
pen or perhaps a slight incompetence on the part of the translator. This is probably
the best interpretation available, but a little more may perhaps be inferred from the
fact that such a misprint found its way into a professional journal — as did a number
of others. First, this fact suggests that conceptual handling of series and series of nu-
merical values was not very well established among the mathematical class to which
CRELLE belonged. And secondly, we may also be tempted to infer something on the
58 “Lehrsatz VI. Bezeichnet man durch ρ0, ρ1, ρ2 u.s.w., ρ ′ 0 , ρ′ 1 , ρ′ 2 u.s.w. die Zahlenwerthe der resp. Glieder
zweier convergenten Reihen
so sind die Reihen
v0 + v1 + v2 + . . . = p und
v ′ 0 + v′ 1 + v′ 2 + . . . = p′ ,
ρ0 + ρ1 + ρ2 + . . . und
ρ ′ 0 + ρ′ 1 + ρ′ 2
ebenfalls noch convergent, und auch die Reihe
deren allgemeines Glied
und deren Summe
+ . . .
r0 + r1 + r2 + · · · + rm
rm = v0v ′ m + v1v ′ m−1 + v2v ′ m−2 + · · · + vmv ′ 0 ,
(v0 + v1 + v2 + . . . ) × � v ′ 0 + v′ 1 + v′ 2
ist, wird convergent seyn.” (N. H. Abel, 1826f, 316–317).
59 (N. H. Abel, 1881, 225).
+ . . . �