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

10.3. Early rigorization of theory of series 199
portantly, GAUSS’ investigations of the hypergeometric series also contained a number
of very interesting results. In particular, GAUSS’ attitude toward convergence of series
and criteria for deciding the convergence is relevant to the present analyses.
Concepts of convergence and a criterion of convergence. Before he could advance
to deeper questions, GAUSS emphasized that the convergence or divergence of the
hypergeometric series had to be investigated. In his paper on the hypergeometric
series, GAUSS gave no explicit definition of convergence. However, by the following
argument, GAUSS claimed that the series converged for |x| < 1 and diverged for |x| >
1. As was customary, these requirements were stated verbally without the notation of
numerical values (see, e.g., quotation below).
GAUSS compared the coefficients of two sequential powers of x, say the coefficients
of x m and x m+1 , and found that their ratio
1 + γ+1
m
1 + α+β
m
+ γ
m 2
+ αβ
m 2
approached the value 1 when m was taken to be increasingly large. GAUSS then con-
cluded that for any complex value of x with |x| < 1, the series would be convergent “at
least from some point onward” and lead to a determinate finite sum. In case |x| > 1,
the series would necessarily diverge and it could not have a sum. 19 GAUSS summa-
rized his position:
“Since our function is defined as the sum of a series, it is obvious, that our
investigations are naturally confined to the cases in which the series actually converges
and that it is absurd to ask for the value of the series whenever x has a
value greater than unity.” 20
Of the cases with |x| = 1, GAUSS only investigated x = 1 and found that under the
condition α + β − γ < 0, the series would have a finite sum. 21
In the above context, GAUSS appears to have employed a concept of series conver-
gence which corresponded to the partial sums approaching a finite limit. We are easily
led to believe that GAUSS’ familiar looking notions such as convergent and sum meant
the same to him as they do to us. However, another concept of convergence was also
in use at GAUSS’ time and even appeared later in his manuscripts (see below). There-
fore, it is worth re-examining the evidence to see if it appears different with this added
information.
Originating with J. LE R. D’ALEMBERT (1717–1783) in the mid-eighteenth century,
the term convergent was used by mathematicians within the formal paradigm to denote
19 (ibid., 126).
20 “Patet itaque, quatenus functio nostra tamquam summa seriei definita sit, disquisitionem natura
sua restrictam esse ad casus eos, ubi series revera convergat, adeoque quaestionem ineptam esse,
quinam sit valor seriei pro valore ipsius x unitate maiori.” (ibid., 126). For a German translation, see
(C. F. Gauss, 1888, 10).
21 (C. F. Gauss, 1813, 139, 142–143).