§2. Absolutely convergent series
133
Theorem 8 has no application outside the domain of absolutely convergent series: the series E l/n diverges and the alternating series E( -1)n /n
converges even though (CP 2) holds in this case. In a case of this kind, one
clearly has to take account of the signs of the terms, or, in the complex case,
of their arguments. The relation
allows one to go a little further; by definition this means (nO 4) that
(16.6) lim Un/Vn = 1, lim vn/un = 1, Un = VnWn with lim Wn = 1,
these three relations clearly being equivalent 41 .
Since a sequence which tends to 1 is bounded and remains bounded away
from 0, this relation implies Un ::=:: Vn .
Theorem 9. Let E Un be a series with terms> 0 and E Vn a series with
complex terms such that Un rv Vn for n large.
(i) If the series E Un converges then the series E Vn converges absolutely;
(ii) If the series E Un diverges, so does the other.
Case (i) follows from (CP 1). In case (ii), consider the series E Wn where
Wn = Re(vn ). Since Re(vn/un ) = wn/un tends to 1 for the same reason as
vn/un , and since Un > 0, one also has Wn > 0 for n large, so the series E Un
and E Wn are simultaneously convergent or divergent. But if E Wn diverges,
so a fortiori does E Vn , qed.
For example, for a series with complex terms,
(16.7)
/
s
{ absolute convergence if s > 1,
Un rv C n =* d'
'f < 1
Ivergence 1 S _ •
Suppose for example that Un = f(n)/g(n) where! and 9 are polynomials of
degrees p and q. One has
_ ap n P (I+?/n + ... +?/n P ) _ p_q 1 + ...
U n -
-en
- - -
bqn q (I+?/n+ ... +?/n q )
1+ ...
with c = ap/bq =f:. 0 and numerical coefficients? whose values do not matter.
Since the fraction on the right hand side tends to 1, it is clear that Un rv en P - q
for n large; hence
'L!(n)/g(n) { converges absolutely if dO (g) 2: dO (f) + 2
diverges if dO(g) :S dOe!) + 1.
Thus the series 2::: series E(n 2 + 3in - 5)/(n 3 - 2i) is divergent.
41 This obviously assumes that, for n large, Un and Vn are never zero. Vain pedantry:
one never meets any other case in practice.