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Lecture 5

MTH 240 Lecture Notes - Lecture 5: Ratio Test, Ibm System P


Department
Mathematics
Course Code
MTH 240
Professor
Changping Wang
Lecture
5

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By Saeid Samiezadeh
This document is intended to summarize various tests used to determine convergence/divergence of series
in Calculus II (MTH 240). It CANNOT be used in anyway as an aid in closed-book tests/quizzes.
Series or Test Form of Series Condition for
Convergence Condition for
Divergence Comments
Geometric Series 0
n
n
ar
0a 1r
1r If convergent,
01
n
n
a
ar r
Useful for comparison tests
Divergence Test 1n
n
a
Does not apply lim 0
n
na

Cannot be used to prove
convergence
Integral Test
1n
n
a
where
()
n
afnand f is
continuous, positive,
and decreasing
1()
f
xdx
converges 1()
f
xdx
diverges
The value of the integral is not
the value of the series.
P-series 1
1
p
nn
1p 1p
Useful for comparison tests
Telescopic Series
1nn
n
ab
whose
partial sums ( n
s)
only have a fixed
number of terms
after cancellation
lim n
ns

 lim n
ns

 The sum of the series can be
found by lim n
ns
 .
Direct
Comparison Test 1n
n
a
where 0
n
a
and 0
n
b
nn
ab
and
1n
n
b
converges
nn
ba
and
1n
n
b
diverges 1n
n
a
is given; you supply
1n
n
b
The test determines absolute
convergence.
Limit
Comparison Test 1n
n
a
where 0
n
a
and 0
n
b
0lim
n
nn
a
b


and
1n
n
b
converges
0lim
n
nn
a
b


and
1n
n
b
diverges
1n
n
a
is given; you supply
1n
n
b
The test determines absolute
convergence.
Alternating
Series Test 1
(1)
nn
n
b
n
bis non-
increasing and
lim 0
n
nb

Does not
apply
If lim 0
n
nb
 , divergence test
shall be used.
Ratio Test 1n
n
a
1
lim 1
n
nn
a
a

1
lim 1
n
nn
a
a
 Inconclusive if 1
lim 1
n
nn
a
a

Root Test 1n
n
a
lim 1
nn
na

lim 1
nn
na
 Inconclusive if
lim 1
nn
na

Absolute
Convergence 1n
n
a
1n
n
a
converges
Does not
apply Applies to arbitrary series
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