(12.4) The Comparison Tests
Often, a series will resemble a simpler series whose convergence is known. This may enable
us to determine if a series converges.
An example is the series
∞
n
=1
1
1 + 2
n
.
We notice that
1
1 + 2
n
≤
1
2
n
, so we know that
∞
n
=1
1
1 + 2
n
≤
∞
n
=1
1
2
n
.
Since
∞
n
=1
1
2
n
= 1, we conclude that
∞
n
=1
1
1 + 2
n
≤
1.
So we are
assured that this series converges. (To what, we don’t know. It converges to some number
≤
1. The partial sum with thirty terms adds up to approximately 0
.
7645.)
The comparison test.
Suppose 0
≤
b
n
≤
a
n
for all
n
, and we know that
∞
n
=1
a
n
is
convergent. Then so is
∞
n
=1
b
n
.
Suppose 0
≤
a
n
≤
b
n
and we know that
∞
n
=1
a
n
is divergent. Then so is
∞
n
=1
b
n
.
Example.
∞
n
=1
n
n
3
+ 1
. Notice
n
n
3
+ 1
≤
n
n
3
=
1
n
2
. Since
∞
n
=1
1
n
2
is convergent, the series
∞
n
=1
n
n
3
+ 1
is also convergent.
Example.
∞
n
=2
2
n
n
2

1
. Notice
2
n
n
2

1
≥
2
n
n
2
=
2
n
≥
1
n
. So
∞
n
=2
2
n
n
2

1
≥
∞
n
=2
1
n
=
∞
. So
the series
∞
n
=2
2
n
n
2

1
diverges.
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 Fall '08
 VALDIMARSSON
 Mathematical Series, lim

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