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

10.3. Early rigorization of theory of series 203
on uncritical use of the binomial theorem
√ 1
−1 = (1 − 2) 2 = 1 − 1 1 1
· 2 − · 4 − · 8 − . . . (10.1)
2 8 16
exhibited the imaginary unit as an infinite sum of real numbers. 31
Following his program, BOLZANO rejected all previous proofs: NEWTON’S proof
because it had been based on extrapolation and not “everything which corresponds
to known truths is necessarily true”; 32 the proofs based on the expansion into Taylor
series because they introduced a vicious circle and the binomial theorem should be the
more basic of the two theorems; and even EULER’S second proof because it operated
with completed infinite series and did not consider the convergence of the series.
Revision of EULER’S proof. Subsequent to his critical remarks, BOLZANO presented
his own new rigorized proof of the binomial theorem. He based it on the outline
of EULER’S second proof but replaced the way in which EULER handed infinite series
with his new concept of numerical equality and limits. Overturning EULER’S manipu-
lations of completed infinite series, BOLZANO worked with the partial sums and limit
arguments. Expressed in EULER’S notation, BOLZANO proved by multiplying the first
s terms of [m] with the first t terms of [n], that the first min (s, t) terms of the product
corresponded to the first min (s, t) terms of [m + n]. 33 If we introduce the notation [m] t
s
to denote the sum of the terms ranging from s to t in the series [m], 34 the result can be
expressed as
�
[m] s
1 [n]t
�min(s,t) 1 = [m + n]
1
min(s,t)
1 .
However, the terms after min (s, t) would not always be equal in the two expressions
but BOLZANO found that the difference
�
[m] s
1 [n]t
�r+s 1
min(s,t)
− [m + n]s+t
min(s,t)
could be made smaller than any given positive value by taking min (s, t) sufficiently
large provided that |x| < 1. 35 Thus, BOLZANO obtained his proof of the functional
equation
under the important assumption |x| < 1.
[m] · [n] = [m + n]
Extension to real exponents. EULER’S (second) proof of the binomial theorem had
focused on rational exponents. At the end of the argument, he suggested that other
(positive) exponents could also be considered:
31 (ibid., vi).
32 (ibid., xi).
33 (ibid., §38).
34 This notation has been adapted from (Hauch, 1997).
35 (Bolzano, 1816, §40).