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

200 Chapter 10. Toward rigorization of analysis
series in which the numerical value of the general term vanished monotonically, i.e.
series ∑ an for which the sequence |an| was monotonically decreasing and approached
zero. 22 The vanishing of terms, clearly contrasted to the convergence of the partial
sums, can be found in an unpublished manuscript written by GAUSS probably after
1831. 23
“By convergence of an infinite series, I will simply understand nothing but the
infinite approaching of its terms toward 0 when the series is infinitely continued.
The convergence of a series in itself is thus to be distinguished from the convergence
of its summation toward a finite limit; however, the latter implies the former
but not the other way around.” 24
Exactly which concept of convergence, GAUSS had in mind in his research on the
hypergeometric series can seem unclear. From a modern perspective, we are tempted
to assume that GAUSS interpreted convergence as convergence of the partial sums and
interpret GAUSS’ comparison of subsequent terms as an implicit quotient criterion.
However, GAUSS’ reasoning can equally well be interpreted within the older concept
of D’ALEMBERT-convergence. 25
In terms of the development described in the next chapter, GAUSS’ investigation
on the hypergeometric series is important in three respects:
1. GAUSS’ investigation was confined to a particular series, albeit one with three
parameters which enabled GAUSS to model a number of transcendental func-
tions using it.
2. GAUSS insisted on establishing the convergence of the series before speaking of
its sum. He used an implicit theorem — apparently equivalent to the ratio test 26
(see subsequent chapters) — to determine restrictions on the variable x.
3. Despite aiming at “the rigorous methods of the ancient geometers” 27 , GAUSS’
theory of infinite series as expressed in the paper on the hypergeometric series
was rudimentary and not spelled out in much detail. For instance, it is not com-
pletely clear precisely what his basic notions meant.
22 (Grabiner, 1981b, 60).
23 (Schneider, 1981, 55–56).
24 “Ich werde unter Convergenz, einer unendlichen Reihe schlechthin beigelegt, nichts anders verstehen
als die beim unendlichen Fortschreiten der Reihe eintretende unendliche Annäherung ihrer
Glieder an die 0. Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz
ihrer Summirung zu einem endlichen Grenzwerthe; letztere schliesst zwar die erstere ein,
aber nicht umgekehrt.” (C. F. Gauss, Fa, Kapsel 46a, A1–A13, 400).
25 In most of the (earlier) secondary literature, e.g. (Pringsheim, 1898–1904, 79), GAUSS’ emphasis on
establishing the convergence of the hypergeometric series and his use of the quotient comparison
have been taken as precursors of the rigorization program (see next chapter). SCHNEIDER has aptly
interpreted GAUSS’ concept of convergence in terms of the sequence of terms (Schneider, 1981, 56).
26 The ratio test is also sometimes called the quotient test but I will use the term ratio test, throughout.
27 “Ostendemus autem, et quidem, in gratiam eorum, qui methodis rigorosis antiquorum geometrarum
favent, omni rigore.” (C. F. Gauss, 1813, 139).