MATH 38
Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
Outline
Summary of Convergence Tests for Infinite Series
These may come in handy...
Remark
To find the common ratio and first term of a geometric series, put it in the
form
n=0
ar n
These may come in handy...
Remark
To find the common ratio and first term of a geometric series, put it in the
form
n=0
Example
1
2n+1
X
n
n=0 3
1
X
n
n=1 2
(1)n+1 5n1
X
n=0
3n 2n+1
ar n
Before we proceed...
True or False
Suppose lim an = 0.
n
Before we proceed...
True or False
Suppose lim an = 0.
n
The sequence {an } is convergent.
Before we proceed...
True or False
Suppose lim an = 0.
n
The sequence {an } is convergent.
The series
a n convergent.
Before we proceed...
True or False
Suppose lim an = 0.
n
The sequence {an } is convergent.
The series
a n convergent.
Remark
Know the difference between a convergent sequence and a convergent
series.
Summary of Convergence Tests for Infinite Series
Section 1.5 of Lecture Notes
List of all convergence tests
Set
Set
Set
Set
A: Test for Divergence
B: Tests for Series of Nonnegative Terms
C: Test for Alternating Series
D: Tests for Absolute Convergence
List of all special series
Some additional tips
Please add:
1
If a series is telescoping,
Please add:
1
If a series is telescoping, find SPS to determine if convergent.
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series,
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
If both are convergent, then
(a n b n )
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
If both are convergent, then
(a n b n ) is convergent.
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
If both are convergent, then
get the sum!)
(a n b n ) is convergent. (Know how to
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
If both are convergent, then (an bn ) is convergent. (Know how to
get the sum!)
P
If one is divergent and the other is convergent, then (an bn )
P
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
If both are convergent, then (an bn ) is convergent. (Know how to
get the sum!)
P
If one is divergent and the other is convergent, then (an bn ) is
divergent.
P
Please add:
1
2
If a series is telescoping, find SPS to determine if convergent.
P
P
P
If a series is of the form (an bn ) where an and bn are special
series, use four convergence theorems:
If both are convergent, then (an bn ) is convergent. (Know how to
get the sum!)
P
If one is divergent and the other is convergent, then (an bn ) is
divergent.
P
If you are asked to get the sum, the series must be telescoping or
geometric (for now).
Convergent or Divergent?
1
Ask first: Is the divergence test applicable? Can it be easily applied?
( lim an is easy to compute and it seems that the limit is not zero)
n
Convergent or Divergent?
1
Ask first: Is the divergence test applicable? Can it be easily applied?
( lim an is easy to compute and it seems that the limit is not zero)
Guess (intelligently)
Convergent or Divergent?
1
Ask first: Is the divergence test applicable? Can it be easily applied?
( lim an is easy to compute and it seems that the limit is not zero)
Guess (intelligently)
Determine the appropriate test to use. Set D Tests may be used
because if the IS is absolutely convergent, then the IS is convergent.
Convergent or Divergent?
1
Ask first: Is the divergence test applicable? Can it be easily applied?
( lim an is easy to compute and it seems that the limit is not zero)
Guess (intelligently)
Determine the appropriate test to use. Set D Tests may be used
because if the IS is absolutely convergent, then the IS is convergent.
NO CONCLUSION is NOT an acceptable final answer.
Example
Determine if the following infinite series are convergent or divergent.
1
p
n +1
p
n=1 2n 2 1
Example
Determine if the following infinite series are convergent or divergent.
1
p
n +1
p
n=1 2n 2 1
2e n n
X
n
n=1 e + 4n
Example
Determine if the following infinite series are convergent or divergent.
1
p
n +1
p
n=1 2n 2 1
2e n n
X
n
n=1 e + 4n
X
1
5n
e
+ n
n
n=1
Example
Determine if the following infinite series are convergent or divergent.
1
p
n +1
p
n=1 2n 2 1
2e n n
X
n
n=1 e + 4n
X
1
5n
e
+ n
n
n=1
X
n=1
3
3
5n 3 5n + 2
Example
Determine if the following infinite series are convergent or divergent.
1
p
n +1
p
n=1 2n 2 1
2e n n
X
n
n=1 e + 4n
X
1
5n
e
+ n
n
n=1
3
3
5n
3
5n
+2
n=1
p
X n
n=2 ln n
Example
Determine if the following infinite series are convergent or divergent.
1
p
n +1
p
n=1 2n 2 1
2e n n
X
n
n=1 e + 4n
X
1
5n
e
+ n
n
n=1
3
3
5n
3
5n
+2
n=1
p
X n
n=2 ln n
X
1
n=0 4
n + 4n
Remark
AST may not always be the best way to show that an alternating series is
convergent.
Remark
AST may not always be the best way to show that an alternating series is
convergent. Sometimes, it is easier to show that a series with negative
terms is convergent via ABSOLUTE CONVERGENCE.
Remark
AST may not always be the best way to show that an alternating series is
convergent. Sometimes, it is easier to show that a series with negative
terms is convergent via ABSOLUTE CONVERGENCE.
Example
Determine whether the following infinite series are convergent or divergent.
1
(1)n (2n + 1)
X
(n + 1)!
n=0
Remark
AST may not always be the best way to show that an alternating series is
convergent. Sometimes, it is easier to show that a series with negative
terms is convergent via ABSOLUTE CONVERGENCE.
Example
Determine whether the following infinite series are convergent or divergent.
1
(1)n (2n + 1)
X
(n + 1)!
n=0
(1)n+1 sin
X
n=1
n2
Absolutely Convergent, Conditionally Convergent or Divergent?
1
Is
a n absolutely convergent?
Absolutely Convergent, Conditionally Convergent or Divergent?
1
Is
a n absolutely convergent?
P
Is |an | convergent? Applicable tests: A, B or D.
Absolutely Convergent, Conditionally Convergent or Divergent?
1
Is
If it is not absolutely convergent and you have not been able to
conclude that it is divergent (via ratio or root test): Is it conditionally
convergent?.
a n absolutely convergent?
P
Is |an | convergent? Applicable tests: A, B or D.
Absolutely Convergent, Conditionally Convergent or Divergent?
1
Is
If it is not absolutely convergent and you have not been able to
conclude that it is divergent (via ratio or root test): Is it conditionally
convergent?.
a n absolutely convergent?
P
Is |an | convergent? Applicable tests: A, B or D.
Is
a n convergent?
Absolutely Convergent, Conditionally Convergent or Divergent?
1
Is
If it is not absolutely convergent and you have not been able to
conclude that it is divergent (via ratio or root test): Is it conditionally
convergent?.
a n absolutely convergent?
P
Is |an | convergent? Applicable tests: A, B or D.
Is
3
a n convergent?
If it is not conditionally convergent, it must be divergent. Justify.
Example
Determine whether the following infinite series are absolutely convergent,
conditionally convergent or divergent.
1
cos n
X
2
n=1 n
(1)n n
X
2
n=1 10n + 1
(1)n n
X
n=1 10n + 1
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Is it absolutely convergent?
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Is it absolutely convergent?
Is
cos n
X
2
n
n=1
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Is it absolutely convergent?
Is
cos n
X
X | cos n|
2 =
n2
n
n=1
n=1
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Is it absolutely convergent?
Is
cos n
X
X | cos n|
convergent?
2 =
n2
n
n=1
n=1
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Is it absolutely convergent?
Is
cos n
X
X | cos n|
convergent? YES
2 =
n2
n
n=1
n=1
Example
Determine whether
cos n
X
is absolutely convergent, conditionally
2
n=1 n
convergent or divergent.
Is it absolutely convergent?
Is
cos n
X
X | cos n|
convergent? YES
2 =
n2
n
n=1
n=1
The series is absolutely convergent.
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely convergent?
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
2
10n + 1
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
convergent?
2
10n + 1
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
convergent? NO
2
10n + 1
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
convergent? NO
2
10n + 1
n=1 10n + 1
Is it conditionally convergent?
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
convergent? NO
2
10n + 1
n=1 10n + 1
Is it conditionally convergent?
Is
(1)n n
X
convergent?
2
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
convergent? NO
2
10n + 1
n=1 10n + 1
Is it conditionally convergent?
Is
(1)n n
X
convergent? YES
2
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
2
n=1 10n + 1
convergent or divergent.
Is it absolutely
(1)n n
X
Is
2
n=1
convergent?
X
n
=
convergent? NO
2
10n + 1
n=1 10n + 1
Is it conditionally convergent?
Is
(1)n n
X
convergent? YES
2
n=1 10n + 1
The series is conditionally convergent.
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely convergent?
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
Is
n=1
10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
convergent?
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
convergent? NO
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
convergent? NO
Is it conditionally convergent?
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
convergent? NO
Is it conditionally convergent?
Is
(1)n n
X
convergent?
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
convergent? NO
Is it conditionally convergent?
Is
(1)n n
X
convergent? NO
n=1 10n + 1
Example
Determine whether
(1)n n
X
is absolutely convergent, conditionally
n=1 10n + 1
convergent or divergent.
Is it absolutely
convergent?
(1)n n
X
X
n
=
Is
n=1
10n + 1
n=1 10n + 1
convergent? NO
Is it conditionally convergent?
Is
(1)n n
X
convergent? NO
n=1 10n + 1
The series is divergent.
Announcement:
Chapter 1 Quiz will be on December 12, Thursday (Lecture Class)
You will be allowed to take home your SW bluebook tomorrow.
Your Chapter 1 SW Bluebook will be collected on Dec 12, during
lecture class.
