Math 617
Mertens's theorem
September 14, 2006
The theorem of Mertens about the Cauchy
product of innite series
P
P1
If the two series 1
a
and
b both converge absolutely, then one
n
n=0
n=0 n
can freely rearrange terms to nd that
1
1
1
n
X
X
X
X
an
cn ;
where cn =
ak bn k :
(1)
bn =
n=0
n=0
n=0
k =0
Franz Carl Joseph Mertens (1840{1927)
observed1 that (1) still holds when
P
only one of the rst
two series, say n an , converges absolutely, as long as
P
the second series n bn converges conditionally. The argument of Mertens
goes as follows.
P
P
Let An , Bn , and Cn denote the partial sums nk=0 ak , nk=0 bk , and
=0 . It suces to prove that both (i) limn!1 (C2n An Bn ) = 0 and
(ii) limn!1 (C2n+1 An+1 Bn ) = 0. For (i), observe that C2n An Bn equals
Proof.
Pn
ck
k
+ bn+2 + + b2n ) + a1 (bn+1 + bn+2 + + b2n 1 ) + + an 1 bn+1
+ an+1 (b0 + b1 + + bn 1 ) + an+2 (b0 + b1 + + bn 2 ) + + a2n b0 : (2)
a0 (bn+1
By hypothesis,
there are numbers A and B such that m
jaj j < A and
j =0
P
m
b < B for all m. Fix a positive . By hypothesis, there exists a
j =0 j
number N such that when n N and m 1, one has
n+m
n+m
X
X
jaj j < A + B and bj <
:
A+B
j =n+1
j =n+1
P
Now (2) shows that when n N , one has that jC2n An Bn j < A A+ B +
B A+
= . Thus limn!1 (C2n An Bn ) = 0 as claimed.
B
To establish the limit (ii), observe that C2n+1 An+1 Bn equals
+ bn+2 + + b2n+1 ) + a1 (bn+1 + bn+2 + + b2n ) + + an bn+1
+ an+2 (b0 + b1 + + bn 1 ) + an+3 (b0 + b1 + + bn 2 ) + + a2n+1 b0 ;
a0 (bn+1
and argue analogously to case (i).
1 F. Mertens, Ueber die Multiplicationsregel f
ur zwei unendliche Reihen,
die Reine und Angewandte Mathematik 79 (1874) 182{184.
Theory of Functions of a Complex Variable I
Journal fur
Dr. Boas