(12.6) Absolute Convergence
We now are considering series whose terms might be positive or negative.
But most of
our convergence tests apply to series whose terms are positive only.
To deal with this,
we deal with the series of absolute values
∞
n
=1

a
n

for a series
∞
n
=1
a
n
. It turns out that if

a
n

converges, so does
a
n
. If

a
n

converges, we say the original series
a
n
is
absolutely convergent
. If
a
n
converges but

a
n

does not converge, we say
a
n
is
conditionally convergent
.
Example.
∞
n
=1
(

1)
n
1
n
2
. The series of absolute values is
∞
n
=1
(

1)
n
1
n
2
=
∞
n
=1
1
n
2
. This is
a
p
series with
p
= 2, so it converges. Therefore the original series
∞
n
=1
(

1)
n
1
n
2
.converges
absolutely.
Example.
∞
n
=1
(

1)
n
1
n
. The series of absolute values is
∞
n
=1
(

1)
n
1
n
=
∞
n
=1
1
n
. This is
the harmonic series (a
p
series with
p
= 1), so it diverges. By the alternating series test,
∞
n
=1
(

1)
n
1
n
converges. Therefore the original series
∞
n
=1
(

1)
n
1
n
2
.converges conditionally.
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 Fall '08
 VALDIMARSSON
 Mathematical Series, lim

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