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absolute convergence of double series∗
pahio†
2013-03-22 2:32:21
Let us consider the double series
∞
X
uij
(1)
i,j=1
of real or complex numbers uij . Denote the row series uk1 +uk2 + . . . by Rk ,
the column series u1k +u2k + . . . by Ck and the diagonal series u11 +u12 +u21 +
u13 +u22 +u31 + . . . by DS. Then one has the
Theorem. All row series, all column series and the diagonal series converge
absolutely and
∞
∞
X
X
Ck = DS,
Rk =
k=1
k=1
if one of the following conditions is true:
• The diagonal series converges absolutely.
• There exists a positive number M such that every finite sum of the numbers |uij | is 5 M .
• The row series Rk converge absolutely and the series W1 +W2 + . . . with
terms
∞
X
|ukj | = Wk
j=1
is convergent. An analogical condition may be formulated for the column
series Ck .
Example. Does the double series
∞ X
∞
X
n−m
m=2 n=3
∗ hAbsoluteConvergenceOfDoubleSeriesi
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converge? If yes, determine its sum.
∞ m
X
1
The column series
have positive terms and are absolutely conn
m=2
verging geometric series having the sum
(1/n)2
1
1
1
=
=
−
= Wn .
1 − 1/n
n(n−1)
n−1 n
The series W3 +W4 + . . . is convergent, since its partial sum is a telescoping sum
N
X
n=3
Wn
N X
1
1
1 1
1 1
1 1
1
1
=
−
=
−
+
−
+
−
+. . .+
−
n−1 n
2 3
3 4
4 5
N −1 N
n=3
equalling simply 21 − N1 and having the limit 12 as N → ∞. Consequently, the
given double series converges and its sum is 21 .
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