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

12.7. From power series to absolute convergence 247
more general theorem in order to illustrate that the requirements of his own theorem
were necessary. However, as F. MERTENS (1840–1927) was later to show, 49 if scru-
tinized more carefully, the example illustrated that the multiplication theorem could
fail if both factors were non-absolutely convergent. This led P. L. WANTZEL (1814–
1848) to prove a version of the multiplication theorem which only assumed absolute
convergence of one of the factors (the other factor just being assumed convergent).
The real beginning of a concept of absolute convergence came with DIRICHLET’S
paper on primes in arithmetic progression which was published in 1837. 50 In that
paper, DIRICHLET introduced a separation of convergent series into two classes based
on the convergence of the series which resulted when the terms were replaced by
the absolute values: Either the series of absolute values remained bounded or it was
unbounded. For series of the first class (absolutely convergent series), DIRICHLET
stated that their convergence and sum remained unaffected if the order of terms was
altered. In particular, in double (and multiple) sums, the order of summation wold
not effect the result. Dirichlet observed that these properties — which were certainly
nice and expected — could fail to hold for series of the second class, and he gave two
examples of what could happen: a convergent series could either become divergent
or alter its sum if its terms were rearranged. 51
For his Habilitation in 1854, RIEMANN presented a paper on the representability of
functions by trigonometric series. 52 The paper is a milestone in the theory of trigono-
metric series and the theory of integrals and also contains interesting remarks on the
concept of absolute and non-absolute convergence. In the historical preface, RIEMANN
outlined the previous developments in the field and claimed that DIRICHLET’S impor-
tant 1829 paper on the convergence of trigonometric series was directly inspired by
DIRICHLET’S discovery of the distinction between absolute and non-absolute conver-
gence. 53 RIEMANN expressed his belief that the prevalence of power series in analysis
was the reason why those concepts had not previously been separated. 54 There are no
obvious traces of the alleged inspiration visible in DIRICHLET’S paper of 1829 but —
as mentioned — the distinction became very explicit in a paper with a different topic
in 1837.
RIEMANN advanced a step beyond DIRICHLET’S observation of the differences be-
tween absolutely and non-absolutely (conditionally) convergent series when he de-
scribed a very simple method by which the partial sums of a conditionally convergent
series could be made to approach any given value by proper rearrangement of the
terms of the series. Central to RIEMANN’S argument was the realization that if a se-
ries ∑ an was conditionally convergent, the series of its positive and negative terms
49 (Mertens, 1875).
50 (G. L. Dirichlet, 1837), see also e.g. (I. Grattan-Guinness, 1970b, 94–95).
51 See also (ibid., 94–95).
52 (B. Riemann, 1854).
53 (ibid., 235) DIRICHLET’S paper is (G. L. Dirichlet, 1829).
54 (B. Riemann, 1854, 235). See below.