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

270 Chapter 13. ABEL and OLIVIER on convergence tests
section 21.3, ABEL’S comments on OLIVIER’S theorem will serve as one among a class
of cases where counter examples were employed for different ends and with differing
confidence in the early nineteenth century.
Central to ABEL’S proof was the inequality
log (1 + x) < x (13.1)
which he claimed was valid for all positive x. For x ≥ 1, it was obvious to ABEL and
he gave no argument. It can be easily obtained by observing that
x − log (1 + x)
is increasing for x ≥ 1 and positive for x = 1. For x < 1, ABEL gave an argument
employing the expansion of the logarithm into power series as
log (1 + x) =
∞
∑
n=1
(−1) n−1
n
x n = x −
∞
∑
n=1
�
1 x
−
2n 2n + 1
�
x 2n .
From this, he observed that the parentheses were always positive which produced the
desired inequality. Here, ABEL thus rearranged the terms of the logarithmic series
without further ado. 15
ABEL employed the inequality (13.1) for x = 1 n
1
n
or written differently
> log
�
1 + 1
�
n
= log
log (1 + n)
log n
<
n + 1
n
to produce
= log (n + 1) − log n,
�
1 + 1
�
.
n log n
Taking logarithms and using the inequality (13.1) again, ABEL obtained
� � �
log (1 + n)
log log (1 + n) − log log n = log
< log 1 +
log n
1
�
<
n log n
ABEL had thus produced the inequality
which when summed from 2 to n gave
log log (1 + n) < log log n + 1
n log n ,
log log (1 + n) < log log 2 +
n
∑
k=2
1
k log k .
1
n log n .
Since the left hand side obviously became infinite for n = ∞, the series on the right
→ 0.
hand side was divergent contradicting OLIVIER’S theorem since nan = 1
log n
ABEL’S conclusion was again remarkably reserved and apparently underplayed,
“The theorem announced in the above citation is thus at fault in this case.” 16
15 For more on the history of absolute convergence, see section 12.7.
16 “Le théorème énoncé dans l’endroit cité est donc en défaut dans ce cas.” (N. H. Abel, 1828a, (400)).