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

236 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
section 12.1, ABEL observed that a common practice for evaluating infinite sums, say
∑ an had been to transform the series into a power series ∑ anx n , obtain an expression
for the sum and then insert x = 1. Commenting on this practice, ABEL wrote,
“This is probably right, but it appears to me that one cannot assume it without
proof; just because
φ (x) = a0 + a1x + a2x 2 + . . .
for all values of x less than 1, it is not thereby said that the same conclusion holds
for x = 1.” 31
In order to illustrate his point of criticism, ABEL remarked that his claim was cer-
tainly to the point if the power series failed to converge for x = 1 in which case it had
no sum. However, when he gave explicit examples, he took them from the emerging
theory of trigonometric series and not from within the realm of power series. And
there is very good reason why he did not give a power series as a counter example;
his fourth theorem states that for power series, the procedure of passing to the limit
(inserting x = 1) can only fail in case the series is divergent for x = 1. Thus, Lehrsatz IV
is the assurance needed to justify this procedure for the class of power series provided
the resulting series is assumed to converge.
DIRICHLET’S modification of ABEL’S proof. In 1862, J. LIOUVILLE (1809–1882) re-
ported having discussed ABEL’S fourth theorem with his friend DIRICHLET, who had
died just a few years before. LIOUVILLE had expressed his concern about the orig-
inal proof of ABEL’S very important theorem which he found difficult to present in
courses and even to understand. On the spot, DIRICHLET gave an alternative proof of
Lehrsatz IV, which LIOUVILLE felt would remove all such difficulties. It was this new
proof by DIRICHLET which LIOUVILLE reproduced in verbatim in a short note in his
Journal de mathématiques pures et appliquées. 32 Similarly, a page in G. F. B. RIEMANN’S
(1826–1866) Nachlass contains his reworking of ABEL’S proof which also supports the
impression that ABEL’S original proof was not universally accepted. 33
Before the differences between the ABEL’S and DIRICHLET’S proofs are discussed
and analyzed, a presentation of DIRICHLET’S new proof is required. A modern recon-
struction of DIRICHLET’S proof is given in box 3.
For the infinite series
A =
∞
∑ am,
m=0
31 “Dette er vel rigtigt; men mig synes at man ikke kan antage det uden Beviis, thi fordi man beviser
at
φ (x) = a0 + a1x + a2x 2 + . . .
for alle Værdier af x som er mindre end 1, saa er det ikke derfor sagt at det samme finder Sted for
x = 1.” (Abel→Holmboe, 1826/01/16. N. H. Abel, 1902a, 17).
32 (G. L. Dirichlet, 1862). The proof is also described in (I. Grattan-Guinness, 1970b, 108).
33 (Laugwitz, 1999, 207).