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

274 Chapter 13. ABEL and OLIVIER on convergence tests
On the other hand, ABEL next turned to a function intimately related to the one
studied above, 19
ψ (n) = φm (n) 1−α
.
1 − α
This time, ABEL’S calculations produced the inequality
ψ (n + 1) − ψ (n) > ψ ′ (n + 1)
corresponding to the fact that ψ ′ was a decreasing function. Consequently, through a
number of calculations, ABEL was led to a series
∞
∑
n=1
1
n log m (n) α+1 ∏ m−1
k=1 logk n
which was convergent if α > 0 and another one (corresponding to α = −1)
which was divergent.
∞
∑
n=1
1
n ∏ m−1
k=1 logk n
A logarithmic test of convergence. These methods of constructing convergent and
divergent series led ABEL to a new test of convergence. The underlying idea of ABEL’S
argument starts from the two series, one convergent and the other divergent, and
compares a given series with these two typical ones. He found by simple arguments
based on the results above, that if
�
log
lim
log m+1 n
1
unn ∏ m−1
k=1 logk n
�
> 1, (13.5)
the series ∑ un was convergent. ABEL’S criterion also indicated, that if the limit in
(13.5) was < 1, the series ∑ un would be divergent. In its polished form, ABEL’S
criterion thus became the following:
Theorem 15 For a series of positive terms ∑ un, the limit
�
1 log un
k = lim
n→∞
d
dn logm �
n
log m+1 n
is considered. If k > 1, the series will be convergent; if k < 1, it will be divergent; and if k = 1,
nothing can be said of the convergence or divergence of the series by this test. ✷
This result was later rediscovered by J. L. F. BERTRAND (1822–1900). 20
19 ABEL actually also denoted this function by φ, but to avoid confusion, I have chosen to label it ψ.
20 (Bertrand, 1842).