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

214 Chapter 11. CAUCHY’s new foundation for analysis
Thus, the limit of f (x) 1 x belonged to the same arbitrarily small interval as the limit
of
f (x+1)
f (x) , and thus the two limits were equal. Now, the ratio test followed by letting
f (n) = un and observing that the limits of n√ un and u n+1
un coincided.
11.5 CAUCHY’s proof of the binomial theorem
CAUCHY agreed with his predecessors in considering the binomial theorem a corner
stone of the calculus. His proof of it relied on and promoted two of his new techniques
and concepts in analysis: those of functional equations and continuous functions. 18 As
EULER had done, CAUCHY considered the functional equation
of which he knew that the binomial
f (m) f (n) = f (m + n) (11.2)
f (m) = (1 + x) m
(11.3)
was a continuous solution for all m provided x was fixed. On the other hand, the
function defined by the infinite series
∞
∑
k=0
� �
m
x
k
k
(11.4)
was also a solution to the functional equation (11.2) under the assumptions that it
converged and m was a rational number.
To demonstrate that the series (11.4) satisfied the functional equation, CAUCHY
had to be able to multiply infinite series. He invented a way of multiplying absolutely
convergent series which rigorously established the convergence of the product. 19 Based
on the argument which EULER had also used, CAUCHY then knew that the series (11.4)
coincided with f (m) for all rational values of m. Therefore, the general equality of
(11.3) and (11.4) would be proved if the series was a continuous function of m. In order
to prove the continuity of the series (11.4), CAUCHY devised and proved a general
theorem to the effect that a convergent sum of continuous functions was always a
continuous function. Later, this theorem would arouse much controversy (see below).
CAUCHY’S way of multiplying infinite series. In the Cours d’analyse, CAUCHY in-
vented a way of multiplying two absolutely convergent series such that the product
would be a new convergent series. As he had done throughout, CAUCHY developed
his theory of infinite series in three steps:
1. Series of real, positive terms (section VI.2)
18 For CAUCHY’S theory of functional equations, see (J. Dhombres, 1992).
19 (A.-L. Cauchy, 1821a, 157).