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

12.8. Product theorems of infinite series 253
standards of the newly established journal, which was hampered by some similar and
less grave misprints in the first years, although its standards of technical printing were
quite high.
ABEL’S proof of Cauchy product theorem followed a path similar to those taken by
CAUCHY in his proofs (see section 11.5). ABEL let pm and p ′ m denote the partial sums
of the factors p and p ′ and wrote
After introducing the notation
he found
2m
∑ rk = pmp
k=0
′ m−1
m+ ∑
k=0
u =
p kv ′ 2m−k
� �� �
=t
+
∞
∑ ρk and u
k=0
′ ∞
= ∑ ρ
k=0
′ k ,
m−1
∑ v2m−kp k=0
′ k
� �� �
=t ′
m−1
t < u ∑ ρ
k=0
′ 2m−k and t′ m−1
′ < u ∑ ρ2m−k k=0
. (12.23)
“without reference to the sign”, i.e. for the numerical values of t and t ′ . ABEL then
employed the Cauchy sequence characterization of convergence (for its prominent posi-
tion in the Abelian framework, see above) to ensure that since the series ∑ ρ k and ∑ ρ ′ k
were convergent, the sums
m−1
∑ ρ
k=0
′ 2m−k and
m−1
∑ ρ2m−k k=0
would both tend to zero as m grew to infinity. Thus, ABEL claimed, by setting m equal
to infinity, the equation (12.23) became
∞
∑ rk =
k=0
�
∞
∑ vk k=0
�
×
�
∞
∑ v
k=0
′ �
k .
At this point, the theorem was proved, but ABEL continued his argument by gen-
eralizing the theorem through the use of power series. ABEL now abandoned the
assumptions that both factors had to be absolutely convergent in favor of the assump-
tion that both the factors and the Cauchy product were (simply) convergent. In his
notes on ABEL’S binomial paper, SYLOW wrote of this generalization: “The theorem
VI is due to Cauchy but the new form which it is given [. . . ] originates with ABEL.” 60
The generalized version of the Cauchy product theorem can thus be stated as follows.
Theorem 13 (Generalized Cauchy product theorem) If the three series
∞
∑ tk, k=0
∞
∑ t
k=0
′ k , and
∞
∑
k=0
∑
tnt
n+m=k
′ m
60 “Le théorème VI est dú à Cauchy, mais la forme nouvelle qu’il a reçue page 226 appartient à Abel.”
(ibid., II, 303).