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

12.3. Convergence 229
12.3 Convergence
While ABEL’S definition and use of infinitesimals were not completely in the line of
CAUCHY’S new rigor, his concept of convergence — and the importance which he at-
tributed to it — closely resembled CAUCHY’S.
“Definition. An arbitrary
v0 + v1 + v2 + · · · + vm etc.
will be called convergent if the sum v0 + v1 + · · · + vm steadily approaches a certain
limit for ever increasing values of m. This limit will be called the sum of the
series. In the contrary case, the series is called divergent and therefore has no sum.
From this definition follows that for a series to converge, it will be necessary and
sufficient that the sum vm + vm+1 + · · · + vm+n steadily approach zero for ever
increasing values of m whatever value n may have.” 23
Just as CAUCHY had done, ABEL quickly related the convergence of a series to the
Cauchy criterion and claimed that it constituted a necessary and sufficient condition
for convergence. As described above, the assertion that convergence followed from
the Cauchy criterion was later realized to be non-trivial, but in the 1820s it was con-
sidered obvious. Although both CAUCHY and ABEL drew the connection between
convergence and the Cauchy criterion, ABEL gave the criterion a much more central
position in his theory of series as will be described below (see page 231).
Immediately following his definition of convergence, ABEL made the rather curious
remark that “in every arbitrary series, the general term vm will approach zero.” 24
Judging from the context, an omission of the word “convergent” must have crept in at
this point. 25
An extended ratio test: Lehrsätze I&II. The first theorem of ABEL’S binomial paper
is most remarkable because of its conceptual contents. Without proof (see a modern-
ized proof in box 2), ABEL observed that for any series of positive terms
23 “Erklärung. Eine beliebige Reihe
∞
∑ ρm
m=0
v0 + v1 + v2 + · · · + vm u.s.w.
soll convergent heißen, wenn, für stets wachsende Werthe von m, die Summe v0 + v1 + · · · + vm sich
immerfort eine gewisse Gränze nähert. Diese Grenze soll Summe der Reihe heißen. Im entgegengesetzten
Falle soll die Reihe divergent heißen, und hat alsdann keine Summe. Aus dieser Erklärung
folgt, daß, wenn eine Reihe convergiren soll, es nothwendig und hinreichend sein wird, daß, für
stets wachsende Werthe von m, die Summe vm + vm+1 + · · · + vm+n sich Null immerfort nähert,
welchen Werth auch n haben mag.” (ibid., 313).
24 “In irgend einer beliebigen Reihe wird also das allgemeine Glied vm sich Null stets nähern.” (ibid.,
313).
25 This omission has therefore also been silently corrected in the French translation found in (N. H.
Abel, 1839; N. H. Abel, 1881).