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

11.5. CAUCHY’s proof of the binomial theorem 217
CAUCHY’S novel definition defines continuity not at a point (as is customary to-
day, see below) but on an entire interval enclosed by two boundary points. 23 Thus,
CAUCHY’S implicit choice of infinitesimal ω such that
| f (x + α) − f (x)| = ω
could seem to be independent of x on the interval and the definition would actually
be of what is now known as a uniformly continuous function. The doubt over the proper
interpretation is introduced by the fact CAUCHY’S use of the symbol ω to designate
the infinitesimal: it “hides” the order in which the limit processes are to be carried out.
CAUCHY and series of functions. In a famous theorem which provided an impor-
tant step in CAUCHY’S proof of the binomial theorem, CAUCHY sought to link the
concepts of convergence and continuity. Since we will discuss the theorem in details
in chapter 12, its entire wording and CAUCHY’S proof of it are reproduced here.
“1st theorem. Whenever the different terms of the series u0 + u1 + u2 + · · · + un +
. . . are functions of one and the same variable x and continuous with respect to this variable
in the neighborhood of a particular value for which the series is convergent, the sum
s of the series is also a continuous function of x in the neighborhood of that particular
value.” 24
As was customary, CAUCHY actually presented the proof before he made the theo-
rem explicit. The proof which he gave proceeded along the following lines. If the sum
is split after n terms
s = sn + rn =
n−1
∑ un +
k=0
∞
∑ un, (11.7)
k=n
the partial sum sn is a polynomial and therefore continuous and the remainder rn can
be made less than any given quantity by the convergence of the series. In consequence,
the difference s (x + α) − s (x) could be made less than any assignable quantity and
the sum was therefore continuous. As we are well aware today, with our common
interpretations of the basic concepts of continuity, limits, and convergence, the theo-
rem is false as stated. In section 14.1.2, its future history through the works of ABEL,
P. L. VON SEIDEL (1821–1896) (and G. G. STOKES (1819–1903)) and CAUCHY again
is outlined to understand how ABEL’S contribution to rigorization was accepted and
interpreted.
riable produit toujours un accroissement infiniment petit de la fonction elle-même.” (A.-L. Cauchy,
1821a, 34–35).
23 See also (Bottazzini, 1990, lxxxi–lxxxiii) and (Giusti, 1984).
24 “1.er Théorème. Lorsque les différens termes de la série (1) sont des fonctions d’une même variable
x, continues par rapport à cette variable dans le voisinage d’une valeur particulière pour laquelle la
série est convergente, la somme s de la série est aussi, dans le voisinage de cette valeur particulière,
fonction continue de x.” (A.-L. Cauchy, 1821a, 131–132).