7.2 Numerical Series and Sequences
99
NOTE: series are not multiplicative like sequences are:
X
a
n
b
n
6
=
X
a
n
X
b
n
,
because
1 +
a
1
b
2
+
a
2
b
2
+
. . . a
n
b
n
6
= (1 +
a
1
+
· · ·
+
a
n
)(1 +
b
2
+
· · ·
+
b
n
)
.
(More cross terms on right.)
Theorem 7.2.7
(Increasing & bounded)
.
If
0
≤
a
n
,
∀
n
, then
∑
a
n
converges iff the
partial sums are bounded.
Proof.
(
⇒
) lim
s
n
exists =
⇒ {
s
n
}
bounded.
(
⇐
)
s
n
=
s
n

1
+
a
n
≥
s
n

1
, so monotone. Then
{
s
n
}
bounded implies
{
s
n
}
convergent
by completeness.
Definition 7.2.8.
∑
a
n
is
absolutely convergent
iff
∑

a
n

converges.
∑
a
n
is
conditionally convergent
iff
∑

a
n

diverges but
∑
a
n
converges.
Example 7.2.3.
1. For a positiveterm series, convergence
≡
absolute convergence.
2.
∑
(

1)
n
2
n
and
∑
(

1)
n
n
!
are absolutely convergent, since
∑
1
2
n
and
∑
1
n
!
are conver
gent.
3.
∑
(

1)
n
n
is conditionally convergent, since the harmonic series diverges.
Most comparison tests actually establish absolute convergence. In a moment, we’ll see
that absolute convergence implies convergence, so this is a stronger result.
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 Fall '11
 Wong
 Mathematical Series, Mathematical analysis

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