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Tests for Convergence
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Tests for Convergence
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The Comparison Test
Theorem
6
.
4
.
1
(The Comparison Test)
Suppose
and
are series with all
and
positive and
for each
.
If
converges, then so does
.
If
diverges, then so does
.
Proof
. [Proof Sketch] The condition of the theorem implies that for any
,
from which each claim follows.
Example
6
.
4
.
2
Consider the series
. For each
we have
Since
converges, Theorem
6.4.1
implies that
also converges.
Example
6
.
4
.
3
Consider the series
. It diverges since for each
we have
and
diverges.
William Stein 2006-03-15