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Special Series with Known Convergence:
“p”series:
The Only Two Types
of (nonpower) series, in this course,
for which we that we can compute
exact values:
Geometric Series:
Telescoping Series:
Typical case:
In General:
Page #1 of 5
1
0
1
if
1,
diverges if
1.
1
n
n
n
n
a
ar
ar
r
r
r
∞
∞

=
=
=
=
<

∑ ∑
(
29
0
(
1)
( )
(1)
(0)
(2)
(1)
m
n
f n
f n
S
f
f
f
f
∞
=
+

⇒
=

+

∑
(3)
(2)
(
1)
(
2)
f
f
f m
f m
+

+



O
( )
(
1)
(
1)
( )
f m
f m
f m
f m
+


+
+

(
29
(
29
0
(
1)
( )
lim
=
lim
(
1)
(0)
m
m
n
m
f n
f n
S
f m
f
∞
→ ∞
=
→ ∞
⇒
+

=
+

∑
(
29
(
29
(
29
(
29
0
0
0
n
0
0
0
(
)
( ) , where
, are positive integers
=
(
)
( )
lim
(
1)
(
2)
(
)
(
)
(
1)
(
1)
n
m
n
n
f n
p
f n
n p
f n
p
f n
f m
f m
f m
p
f n
f n
f n
p
∞
=
∞
→ ∞
=
+

+

=
+
+
+
+
+
+

+
+
+
+
+

∑
∑
K
K
(
29
(
29
0
0
0
=
lim
( )
(
)
(
1)
(
1)
m
p
f m
f n
f n
f n
p
→ ∞

+
+
+
+
+

K
1
converges if
1
and
diverges if
1
p
p
p
n
≤
∑
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View Full Document Convergence Tests:
Divergence Test:
Alternating Series Test (AST):
(
But:
If the limit is nonzero, this is the condition for the Divergence Test
, and the series
will diverge
)
The Following Four
Tests
are for use on NonNegative series.
Each can be used on the positive form
of the
series,
, to demonstrate
Absolute Convergence.
1)
Comparison Test:
2)
Limit Comparison Test:
Page #2 of 5
lim
0
then
diverges
(Notice, this test CANNOT show
converges!)
n
n
n
n
a
a
a
→ ∞
≠
∑ ∑
and
converges
converges
diverges
no information
diverges
diverges
converges
no inf
n
n
n
n
n
n
n
n
n
n
n
n
Given
a
b non
negative series
b
a
a
b
b
b
a
a
b
b

⇒
≤
⇒
⇒
⇒
≥
⇒
⇒
∑ ∑
∑ ∑
∑
∑ ∑
∑
ormation
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This note was uploaded on 12/14/2010 for the course MATH 1za3 taught by Professor Ben during the Winter '10 term at Macalester.
 Winter '10
 BEN
 Geometric Series

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