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

12.9. ABEL’s proof of the binomial theorem 259
This characterization contained the complete two-part answer to the questions which
ABEL had raised: the sum of the binomial series when it is convergent and the condi-
tions of its convergence. The cumbersome form of the sum of the binomial series arises
partly from the fact that ABEL expressed its complex variables separated into real and
imaginary parts, and partly from the answer it gives to the problem of multivalued
answers: ABEL’S expression for the sum of the series only has a single value because
the bracket is a positive number and the extraction of roots of positive numbers results
in a canonical, positive value.
An example relating to ABEL’S “exception”. At the very end of the paper, ABEL
used the results which he had found to carry out the summation of certain interesting
series. In particular, the first example is of interest in connection with ABEL’S famous
exception.
In the first example, 62 ABEL proposed to sum the series
α sin φ − 1
2 α2 sin 2φ + 1
3 α3 sin 3φ + . . .
which he found was convergent for |α| < 1 where it converged toward the value β
above,
β = arctan
α sin φ
1 + α cos φ =
∞
∑
n=1
(−1) n−1 sin nφ
α
n
n .
To determine the value for α = 1, it sufficed to let α approach the limit 1 provided the
resulting series remained convergent (Lehrsatz IV). Thus, for φ between −π and π,
1
φ = arctan
2
sin φ
1 + cos φ = ∑ (−1)n−1 sin nφ
.
n
For φ = ±π, the situation was different because the series vanished and the expression
for β degenerated. ABEL observed:
“It follows, that the function
sin φ − 1 1
sin 2φ + sin 3φ − . . .
2 3
has the remarkable property of being discontinuous for the values φ = π and
φ = −π.” 63
Thus, in this case, ABEL used the same object as in the exception for another purpose.
This time, he wanted to illustrate the same point as in the notebook (see section 12.6):
that although the series of the form ∑ vm (x) α m was continuous for α < 1, it needed
not be continuous for α = 1.
62 (ibid., 336–337).
63 “Hieraus folgt, daß die Function:
sin φ − 1 1
sin 2φ + sin 3φ − u.s.w.
2 3
die merkwürdige Eigenschaft hat, für die Werthe φ = π und φ = −π unstetig zu seyn.” (ibid.,
336–337).