11.6
Absolute Convergence and the Ratio and Root Tests
Definition
A series
is called
absolutely convergent
if the series of absolute values
n
a
∑
n
a
∑
is
convergent.
If the series
has only positive terms then
∑
.
n
a
∑


n
n
a
a
=
∑
We have seen that the alternating harmonic series
1
1
( 1)
n
n
n
−
∞
=
−
∑
is convergent. However, it is not
absolutely convergent because
1
1
1
( 1)
1
n
n
n
n
n
−
∞
∞
=
=
−
=
∑
∑
is the harmonic series and is divergent. So
we would say that
1
1
( 1)
n
n
n
−
∞
=
−
∑
is
conditionally convergent
because it is convergent but not
absolutely convergent.
n
a
∑
is
conditionally convergent
if it is convergent but not absolutely convergent.
Theorem:
If a series
is absolutely convergent, then it is convergent.
n
a
∑
Example:
Determine whether the series is absolutely convergent, conditionally convergent or
divergent.
1
( 1)
n
n
n
∞
=
−
∑
.
Note: First check for absolute convergence since that would mean it is convergent. If it
is not absolutely convergence then check for conditional convergence.
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 Fall '08
 Estrada
 Mathematical Series, lim, ∑a

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