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

11.5. CAUCHY’s proof of the binomial theorem 215
2. Series of general real terms (section VI.3)
3. Series of complex terms (chapter IX)
In each of these three theories, CAUCHY developed a product theorem, 20 and CAUCHY’S
proofs will be relevant when compared with ABEL’S subsequent proofs. The multipli-
cation theorems applying to series of real terms (the first and second) are the most in-
teresting for the present study. In his theory of series of positive terms, CAUCHY had
stated and proved that if two series ∑ un and ∑ vn were convergent and converged
toward s and s ′ , the series whose general term was
wk = ∑
n+m=k
unvm
(11.5)
would be convergent and converge toward the product ss ′ . When he wanted to gener-
alize this theorem to general series of real terms, CAUCHY imposed the restriction that
each of the series ∑ un and ∑ vn was to be convergent when their terms were replaced
by their absolute values, i.e. both factors were to be absolutely convergent, although the
term and an elaborate concept was only invented some years later (see section 12.7).
In the first case, in which all terms were positive quantities, CAUCHY proved the
theorem by a direct argument. He let s ′′
n designate the sum of the first n terms of the
purported product series (11.5) and defined
to obtain
He had thus obtained
⎧
⎪⎨
n − 1
if n is odd
m = 2
⎪⎩ n − 2
if n is even
2
n−1
∑ wk <
k=0
n−1
∑ wk >
k=0
�
n−1
∑ uk k=0
�
m
∑ uk k=0
sm+1s ′ m+1
� � n−1
∑
vk k=0
� �
m
∑ vk k=0
�
�
< s′′
n < sns ′ n,
.
and
and by letting m grow beyond all bounds, the theorem was established.
When he came to generalize this theorem to the case of general real terms, CAUCHY
wanted to apply the simpler case of series with positive terms. With the same notation
as above, he obtained the formula
20 (ibid., 141–142,147–149,283–285).
s ′ nsn − s ′′
2n−2
n =
∑
t=n
∑
m+k=t
umv k, (11.6)