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MATH 38
Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
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Outline
1 Summary of Convergence Tests for Infinite Series
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These may come in handy...
Remark
To find the common ratio and first term of a geometric series, put it in theform ∞
n =0ar
n
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These may come in handy...
Remark
To find the common ratio and first term of a geometric series, put it in theform ∞
n =0ar
n
Example
1
∞n =0
2n +1
3n
2 ∞n =1
12n
3
∞n =0
(−1)n +15n −13n 2n +1
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Before we proceed...
True or False
Suppose limn →∞a n = 0.
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Before we proceed...
True or False
Suppose limn →∞a n = 0.
1 The sequence {a n } is convergent.
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Before we proceed...
True or False
Suppose limn →∞a n = 0.
1 The sequence {a n } is convergent.
2
The series a
n convergent.
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Before we proceed...
True or False
Suppose limn →∞a n = 0.
1 The sequence {a n } is convergent.
2
The series a
n convergent.
Remark
Know the difference between a convergent sequence and a convergent
series.
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Summary of Convergence Tests for Infinite Series
Section 1.5 of Lecture Notes
List of all convergence tests
Set A: Test for Divergence
Set B: Tests for Series of Nonnegative TermsSet C: Test for Alternating SeriesSet D: Tests for Absolute Convergence
List of all special series
Some additional tips
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Please add:
1 If a series is telescoping,
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series,
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
If both are convergent, then (a n ±b n )
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
If both are convergent, then (a n ±b n ) is convergent.
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
If both are convergent, then (a n ±b n ) is convergent. (Know how to
get the sum!)
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
If both are convergent, then
(a n ±b n ) is convergent. (Know how to
get the sum!)If one is divergent and the other is convergent, then
(a n ±b n )
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
If both are convergent, then
(a n ±b n ) is convergent. (Know how to
get the sum!)If one is divergent and the other is convergent, then
(a n ±b n ) is
divergent.
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Please add:
1 If a series is telescoping, find SPS to determine if convergent.2 If a series is of the form
(a n ±b n ) where
a n and
b n are special
series, use four convergence theorems:
If both are convergent, then
(a n ±b n ) is convergent. (Know how to
get the sum!)If one is divergent and the other is convergent, then
(a n ±b n ) is
divergent.
3 If you are asked to get the sum, the series must be telescoping orgeometric (for now).
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Convergent or Divergent?1 Ask first: Is the divergence test applicable? Can it be easily applied?
( limn →∞a n is easy to compute and it seems that the limit is not zero)
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Convergent or Divergent?1 Ask first: Is the divergence test applicable? Can it be easily applied?
( limn →∞a n is easy to compute and it seems that the limit is not zero)
2 Guess (intelligently)
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Convergent or Divergent?1 Ask first: Is the divergence test applicable? Can it be easily applied?
( limn →∞a n is easy to compute and it seems that the limit is not zero)
2 Guess (intelligently)
3 Determine the appropriate test to use. Set D Tests may be usedbecause if the IS is absolutely convergent, then the IS is convergent.
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Convergent or Divergent?1 Ask first: Is the divergence test applicable? Can it be easily applied?
( limn →∞a n is easy to compute and it seems that the limit is not zero)
2 Guess (intelligently)
3 Determine the appropriate test to use. Set D Tests may be usedbecause if the IS is absolutely convergent, then the IS is convergent.
4 NO CONCLUSION is NOT an acceptable final answer.
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Example
Determine if the following infinite series are convergent or divergent.
1
∞n =1
n +1
2n 2−1
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Example
Determine if the following infinite series are convergent or divergent.
1
∞n =1
n +1
2n 2−1
2 ∞n =1
2e
n
e n +4n
n
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Example
Determine if the following infinite series are convergent or divergent.
1
∞n =1
n +1
2n 2−1
2 ∞n =1
2e
n
e n +4n
n
3
∞n =1
e −5n + 1
n n
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Example
Determine if the following infinite series are convergent or divergent.
1
∞n =1
n +1
2n 2−1
2 ∞n =1
2e
n
e n +4n
n
3
∞n =1
e −5n + 1
n n
4
∞n =1
3
5n −3− 3
5n +2
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Example
Determine if the following infinite series are convergent or divergent.
1
∞n =1
n +1
2n 2−1
2 ∞n =1
2e
n
e n +4n
n
3
∞n =1
e −5n + 1
n n
4
∞n =1
3
5n −3− 3
5n +2
5 ∞n =2
n lnn
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Example
Determine if the following infinite series are convergent or divergent.
1
∞n =1
n +1
2n 2−1
2 ∞n =1
2e
n
e n +4n
n
3
∞n =1
e −5n + 1
n n
4
∞n =1
3
5n −3− 3
5n +2
5 ∞n =2
n lnn
6
∞n =0
1
4n +4n
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Remark
AST may not always be the best way to show that an alternating series isconvergent.
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Remark
AST may not always be the best way to show that an alternating series isconvergent. Sometimes, it is easier to show that a series with negativeterms is convergent via ABSOLUTE CONVERGENCE.
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Remark
AST may not always be the best way to show that an alternating series isconvergent. Sometimes, it is easier to show that a series with negativeterms is convergent via ABSOLUTE CONVERGENCE.
Example
Determine whether the following infinite series are convergent or divergent.
1
∞n =0
(−1)n (2n +1)
(n +1)!
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Remark
AST may not always be the best way to show that an alternating series isconvergent. Sometimes, it is easier to show that a series with negativeterms is convergent via ABSOLUTE CONVERGENCE.
Example
Determine whether the following infinite series are convergent or divergent.
1
∞n =0
(−1)n (2n +1)
(n +1)! 2
∞n =1
(−1)n +1 sinπ
n
n 2
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Is
a n absolutely convergent?
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Is
a n absolutely convergent?
Is |a n | convergent? Applicable tests: A, B or D.
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Is
a n absolutely convergent?
Is |a n | convergent? Applicable tests: A, B or D.
2 If it is not absolutely convergent and you have not been able toconclude that it is divergent (via ratio or root test): Is it conditionallyconvergent?.
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Is
a n absolutely convergent?
Is |a n | convergent? Applicable tests: A, B or D.
2 If it is not absolutely convergent and you have not been able toconclude that it is divergent (via ratio or root test): Is it conditionallyconvergent?.
Is
a n convergent?
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Is
a n absolutely convergent?
Is |a n | convergent? Applicable tests: A, B or D.
2 If it is not absolutely convergent and you have not been able toconclude that it is divergent (via ratio or root test): Is it conditionallyconvergent?.
Is
a n convergent?
3 If it is not conditionally convergent, it must be divergent. Justify.
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Example
Determine whether the following infinite series are absolutely convergent,conditionally convergent or divergent.
1
∞n =1
cosn
n 2
2∞n =1
(−1)n n 10n 2+1
3
∞n =1
(−1)n n 10n +1
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is∞n =1
cosn n 2
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is∞n =1
cosn n 2
= ∞n =1
|cosn |n 2
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is∞n =1
cosn n 2
= ∞n =1
|cosn |n 2
convergent?
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is∞n =1
cosn n 2
= ∞n =1
|cosn |n 2
convergent? YES
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Example
Determine whether∞n =1
cosn
n 2 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is∞n =1
cosn n 2
= ∞n =1
|cosn |n 2
convergent? YES
∴ The series is absolutely convergent.
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Example
Determine whether ∞n =1
(−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
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Example
Determine whether ∞n =1
(−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
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Example
Determine whether
∞n =1
(−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n
n 10n 2+1
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Example
Determine whether
∞n =1
(−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n
n 10n 2+1
= ∞n =1
n 10n 2+1
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Example
Determine whether
∞n =1
(
−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n
n 10n 2+1
= ∞n =1
n 10n 2+1
convergent?
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Example
Determine whether
∞n =1
(
−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n
n 10n 2+1
= ∞n =1
n 10n 2+1
convergent? NO
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Example
Determine whether
∞n =1
(
−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n
n 10n 2+1
= ∞n =1
n 10n 2+1
convergent? NO
Is it conditionally convergent?
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Example
Determine whether
∞n =1
(
−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n 2+1
= ∞n =1
n
10n 2+1convergent? NO
Is it conditionally convergent?
Is∞n
=1
(−1)n n 10n 2
+1
convergent?
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Example
Determine whether
∞n =1
(
−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n 2+1
= ∞n =1
n
10n 2+1convergent? NO
Is it conditionally convergent?
Is∞n
=1
(−1)n n 10n 2
+1
convergent? YES
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Example
Determine whether
∞n =1
(
−1)n n
10n 2+1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n 2+1
= ∞n =1
n
10n 2+1convergent? NO
Is it conditionally convergent?
Is∞n
=1
(−1)n n 10n 2
+1
convergent? YES
∴ The series is conditionally convergent.
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionally
convergent or divergent.
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionally
convergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1convergent?
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionallyconvergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1convergent? NO
l
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionallyconvergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1convergent? NO
Is it conditionally convergent?
E l
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionallyconvergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1convergent? NO
Is it conditionally convergent?
Is∞n
=1
(−1)n n 10n
+1
convergent?
E l
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionallyconvergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1convergent? NO
Is it conditionally convergent?
Is∞n
=1
(−1)n n 10n
+1
convergent? NO
E l
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Example
Determine whether
∞n =1
(
−1)n n
10n +1 is absolutely convergent, conditionallyconvergent or divergent.
Is it absolutely convergent?
Is ∞n =1
(−1)n n
10n +1
= ∞n =1
n
10n +1convergent? NO
Is it conditionally convergent?
Is∞n
=1
(−1)n n 10n
+1
convergent? NO
∴ The series is divergent.