132
II - Convergence: Discrete variables
16 - Comparison relations. Criteria of Cauchy and d'Alembert
Given two scalar sequences (un) and (vn ), one says that the first is dominated
by the second if there exists a number M ~ 0 such that IUn I ~ M.lvn I for n
large, which one can express by writing (end of nO 3)
(16.1)
when n ~ +00.
On the other hand one says (nO 3, (ii)) that the sequences (un) and
(vn ) are of the same order of magnitude at infinity if both Un = O(vn ) and
Vn = O(un ), which one writes
(16.2)
when n ~ +00.
This means that there exist numbers m > 0 and M > 0 such that
(16.3)
for n large.
For n large, the two ratios un/vn and vn/un thus should stay off 0, as sailors
would say.
These definitions will provide us with comparison principles between series, quasi-trivial it is true, but very useful all the same:
Theorem 8. Let L Un and L Vn be two series with complex terms.
(CP 1) If Un = O(vn ) and if the series L Vn is absolutely convergent, then
so is the series L Un.
(CP 2) If Un ;:0::: Vn and if one of the two series is absolutely convergent, then
so is the other.
If indeed lunl ~ M.lvnl for n large, and if the partial sums of the series L Ivnl
are bounded, then so clearly are those of the series L lunl, whence (CP 1).
The second part follows from the first on exchanging the roles of two series.
Corollary. For a series L Un to be absolutely convergent, it suffices that
there exists a number q such that
o ~ q < 1,
or that there exists a number s such that
s> 1.
In the case of series with positive terms one can go a little further:
(16.4)
if Un ;:0::: qn,
(16.5) if Un ;:0::: l/n s ,
This follows from (CP 2).
the series converges if q < 1 and diverges if q ~ 1;
the series converges if s > 1 and diverges if s ~ 1.