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

226 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
which is the Fourier series corresponding to the function f (x) = x 2 on the interval
]−π, π[. However, as ABEL noted in the letter, by inserting e.g. x = π he would be led
to the absurd equality
π
2 =
∞ (−1)
∑
n=1
n−1 sin nπ
= 0.
n
In his letter, ABEL applied this example to illustrate that although an equality held
for x < π it could fail in the limit x = π. This criticism is thus an elaboration of
CAUCHY’S dismissal of the generality of algebra. ABEL continued,
“Operations are applied to infinite series as if they were finite but is that permissible?
I doubt it. — Where has it been proved that one obtains the differential
of a series by differentiating each term? It is easy to present examples for which
this it not true.” 19
Here, ABEL indirectly criticized J. B. J. FOURIER’S (1768–1830) interchange of the limit
processes involved in term-wise integration. Differentiating the series (12.4), ABEL
obtained
1
2 =
∞
∑ (−1)
n=1
n−1 cos nx
in which the series was divergent. Now, the example (12.3) can be seen to result if this
procedure of differentiation is repeated and either x = π or x = π 2
is inserted.
ABEL’S reaction to Poisson’s example. A strong connection between ABEL’S research
on the theory of series and Poisson’s example is clearly discernible from his letter to
HOLMBOE. 20 There, ABEL explained how he had undertaken to find the sum of the
series
m (m − 1)
cos mx + m cos (m − 2) x + cos (m − 4) x + . . .
2
which was an important open problem at the time. ABEL mentioned that a large num-
ber of mathematicians (including S.-D. POISSON (1781–1840) and CRELLE) had failed
to solve the problem but that he, himself, had found a complete answer in the form
m (m − 1)
cos mx + m cos (m − 2) x + cos (m − 4) x + · · · = (2 + 2 cos 2x)
2
m 2 cos mkπ
�
in which m > −1, k an integer and k − 1 � �
2 π < x < k + 1 �
2 π; for m < −1, the series
was divergent and this led to ABEL’S outburst against the use of divergent series:
“Divergent series are the creations of the Devil and it is a shame that anybody
dare construct a demonstration upon them.” 21
19 “Man anvender alle Operationer paa uendelige Rækker som om de vare endelige, men er dette
tilladt? Vel neppe. — Hvor staar det beviist at man faaer Differentialet af en uendelig Række ved at
differentiere hvert Led? Det er let at anføre Exempler hvor dette ikke er rigtigt.” (Abel→Holmboe,
1826/01/16. N. H. Abel, 1902a, 18).
20 (Abel→Holmboe, 1826/01/16. In ibid., 13–19).