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

246 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
12.7 From power series to absolute convergence
As indicated in his letter to HANSTEEN and in the general approach of the binomial
paper, ABEL put a lot of emphasis on power series in his attempt to rebuild the theory
of series. In particular, ABEL’S replacement for the invalidated Cauchy Theorem was
based on a particular kind of series which were power series in one variable with
coefficients which were continuous functions of another variable. Well into the second
half of the nineteenth century, this particular argument was found to be better recast
within a concept of absolute convergence which had emerged over the century.
Emergence of a concept of absolute convergence. During the 19 th century, numer-
ical (absolute) values of real numbers and the moduli of complex numbers entered
ever more explicitly in arguments of analysis. As described in the examples from
CAUCHY’S Cours d’analyse and ABEL’S binomial paper, the only way mathematicians
could describe numerical values in the first decades of the century was through verbal
formulations. Generally, CAUCHY was quite careful about these in stating his theo-
rems on series; but proper concern for numerical values was often lacking in ABEL’S
formulations. 45 It appears that notation such as |x| was only invented by WEIER-
STRASS in unpublished papers of the 1840s and did not become customary until the
1870s. 46
In the first decades of the 19 th century, series of numerical values mostly entered
the picture in connection with the multiplication theorem. In the Cours d’analyse, when
CAUCHY generalized his multiplication theorem for series of positive terms to more
arbitrary series, he based his argument on the assumption of convergence of the series
of absolute terms. However, despite its use in proving theorems, CAUCHY’S implicit
concept of absolute convergence still lacked most of its later structural position.
Immediately following the proof of the multiplication theorem, CAUCHY took an
interesting step in investigating the consequences of relaxing the assumptions. He
proved, based on squaring the series
∞
∑
n=1
(−1) n−1
√ , (12.18)
n
that the assumption of absolute convergence was indeed necessary: Because the terms
of the alternating series (12.18) are decreasing in absolute value, the series was conver-
gent as CAUCHY had proved. 47 However, it was not absolutely convergent and when
CAUCHY produced the square of the series, he obtained another divergent series. 48
Here, CAUCHY took a rather modern step of using a counter example to a fictitious
45 See e.g. the editors’ remark in (Lakatos, 1976, 134).
46 (K. Weierstrass, 1876, 78) and e.g. (K. Weierstrass, [1841] 1894).
47 (A.-L. Cauchy, 1821a, 144).
48 (ibid., 149–150). The divergence of the series ∑ 1
√ n can be obtained by comparing with the harmonic
series.