SUMMARY OF INFINITE SERIES TESTS
1.
Definition of the sum of a series.
Let
s
n
=
a
1
+
a
2
+
· · ·
+
a
n
. If lim
n
→∞
s
n
exists and equals
s
then
∑
∞
n
=1
a
n
is convergent
and its sum is
s
.
2.
Well understood series.
•
(
geometric series
) The series
a
+
ar
+
ar
2
+
ar
3
+
· · ·
is convergent if

r

<
1 and the
sum is
a/
(1

r
) and the series is divergent with no sum if

r
 ≥
1.
•
(
p series
) The series
∑
∞
n
=1
1
n
p
is convergent if
p >
1 and divergent if
p
≤
1.
•
(
Telescoping series
) For some series
∑
∞
n
=1
a
n
, a partial fraction decomposition of
a
n
results in a telescoping sum of
s
n
making it easy to determine the sum of the series.
3.
Tests for positive term series only.
Suppose
∑
∞
n
=1
a
n
and
∑
∞
n
=1
b
n
are positive term
series.
•
(
limit comparison test
) If
a
n
∼
b
n
for large
n
, then
∑
∞
n
=1
a
n
and
∑
∞
n
=1
b
n
have the
same behavior.
•
(
comparison test
) Suppose
a
n
≤
b
n
for all
n
.
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 Fall '08
 STAFF
 Calculus, Geometric Series, Infinite Series, Mathematical Series, series A, positive term series

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