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

12.6. A curious reaction: Lehrsatz V 245
Assuming that x1 was a value such that x < x1 < α, ABEL introduced a bound by
assuming that the m’th term was the maximum of these differences,
(φm (β − ω) − Am) x m 1
= max
n≥0 {(φn (β − ω) − An) x n 1 } . (12.17)
This step resembles the introduction of the problematic quantity θ (x) in the bino-
mial paper and the existence (i.e. finiteness) of such a maximum was apparently un-
problematic to ABEL. Accordingly, LIE has suggested the same method of saving
ABEL’S argument as SYLOW had done for the Lehrsatz V, i.e. by turning its existence
into an explicit assumption (see above). ABEL concluded that
f (β − ω) − R = ξ
1 − x x 1
(φm (β − ω) − Am) x m 1
for some ξ ∈ [−1, 1]. When he let ω vanish, ABEL observed that the term
φm (β − ω) − Am
also vanished by the continuity of φm. Therefore, ABEL concluded, the function f was
continuous.
As described, ABEL’S two notebook proofs of the Lehrsatz V are slightly different
from the printed version. However, they share the same structure and many of the
methods which they apply, in particular concerning the belief in the existence of uni-
form bounds (12.15 and 12.17). It is tempting to speculate with LIE that ABEL had
realized that his original proof of Lehrsatz V was problematic — perhaps seizing on
the same objection as SYLOW did and proposing the solution which amounts to uni-
form convergence (see above). However, despite the new proofs, ABEL’S treatment
of Lehrsatz V continued to suffer from essentially the same problems and such an in-
terpretation is not compelling. If ABEL had become uneasy about his proof, it was
probably for another reason or perhaps he just wanted another proof of a well estab-
lished result?
Probing the extent of Lehrsatz V. Following his new proof of Lehrsatz V in the note-
book, ABEL observed that the theorem demonstrated the continuity of the function
f (y) =
∞
x
∑
n=1
n sin ny
n
for all x < 1, although for x = 1, the function — which was the “exception” of his
binomial paper — had certain discontinuities. Under similar assumptions, the series
corresponding to x = 1 could also fail to be divergent, altogether, ABEL observed and
exemplified. These remarks again illustrate ABEL’S repeated criticism of the unwar-
ranted passage to the limit in series.