m38 lec 010714
TRANSCRIPT
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MATH 38Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
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Outline
1 Absolute Convergence, Conditional Convergence and Divergence
2 Power Series
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Test for absolutely convergence.
2 If it is not absolutely convergent, test for conditional convergence.3 If it is not conditionally convergent, it is divergent.
RemarkIf the infinite series is not absolutely convergent, it does not mean it isdivergent!!!
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Test for absolutely convergence.2 If it is not absolutely convergent, test for conditional convergence.
3 If it is not conditionally convergent, it is divergent.
RemarkIf the infinite series is not absolutely convergent, it does not mean it isdivergent!!!
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Test for absolutely convergence.2 If it is not absolutely convergent, test for conditional convergence.3 If it is not conditionally convergent, it is divergent.
RemarkIf the infinite series is not absolutely convergent, it does not mean it isdivergent!!!
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Absolutely Convergent, Conditionally Convergent or Divergent?1 Test for absolutely convergence.2 If it is not absolutely convergent, test for conditional convergence.3 If it is not conditionally convergent, it is divergent.
RemarkIf the infinite series is not absolutely convergent, it does not mean it isdivergent!!!
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ExampleDetermine whether the given series is absolutely convergent, conditionallyconvergent or divergent.
1n=1
(1)n(2n+1)n2
2n=1
(1)n3n+1nn
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Outline
1 Absolute Convergence, Conditional Convergence and Divergence
2 Power Series
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DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NotationFor compactness, we write a power series in xa as:
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...=n=0
cn(xa)n
(Again, the lower index may be some other nonnegative integer k.)
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DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NotationFor compactness, we write a power series in xa as:
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...=n=0
cn(xa)n
(Again, the lower index may be some other nonnegative integer k.)
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DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NotationFor compactness, we write a power series in xa as:
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...=
n=0
cn(xa)n
(Again, the lower index may be some other nonnegative integer k.)
-
DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NotationFor compactness, we write a power series in xa as:
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...=n=0
cn(xa)n
(Again, the lower index may be some other nonnegative integer k.)
-
DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NotationFor compactness, we write a power series in xa as:
c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...=n=0
cn(xa)n
(Again, the lower index may be some other nonnegative integer k.)
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Examplen=0
(x2)n2n
is a power series in x2:
n=0
(x2)n2n
= 1+ (x2)2
+ (x2)2
8+ (x2)
3
16+ ...
Examplen=0
xn
n!is a power series in x0 (or simply x):
n=0
xn
n!= 1+x+ 1
2x2+ 1
6x3+ ...
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Examplen=0
(x2)n2n
is a power series in x2:
n=0
(x2)n2n
=
1+ (x2)2
+ (x2)2
8+ (x2)
3
16+ ...
Examplen=0
xn
n!is a power series in x0 (or simply x):
n=0
xn
n!= 1+x+ 1
2x2+ 1
6x3+ ...
-
Examplen=0
(x2)n2n
is a power series in x2:
n=0
(x2)n2n
= 1+ (x2)2
+ (x2)2
8+ (x2)
3
16+ ...
Examplen=0
xn
n!is a power series in x0 (or simply x):
n=0
xn
n!= 1+x+ 1
2x2+ 1
6x3+ ...
-
Examplen=0
(x2)n2n
is a power series in x2:
n=0
(x2)n2n
= 1+ (x2)2
+ (x2)2
8+ (x2)
3
16+ ...
Examplen=0
xn
n!is a power series in x0 (or simply x):
n=0
xn
n!= 1+x+ 1
2x2+ 1
6x3+ ...
-
Examplen=0
(x2)n2n
is a power series in x2:
n=0
(x2)n2n
= 1+ (x2)2
+ (x2)2
8+ (x2)
3
16+ ...
Examplen=0
xn
n!is a power series in x0 (or simply x):
n=0
xn
n!=
1+x+ 12x2+ 1
6x3+ ...
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Examplen=0
(x2)n2n
is a power series in x2:
n=0
(x2)n2n
= 1+ (x2)2
+ (x2)2
8+ (x2)
3
16+ ...
Examplen=0
xn
n!is a power series in x0 (or simply x):
n=0
xn
n!= 1+x+ 1
2x2+ 1
6x3+ ...
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Just a note
When we expand:
n=0
cn(xa)n = c0(xa)0+ c1(xa)1+ c2(xa)2+ ...+ cn(xa)n + ...
= c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NoteIn power series notation:
(xa)0 = 1for all x R (including x = a)
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Just a note
When we expand:
n=0
cn(xa)n = c0(xa)0+ c1(xa)1+ c2(xa)2+ ...+ cn(xa)n + ...
= c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NoteIn power series notation:
(xa)0 = 1for all x R (including x = a)
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Just a note
When we expand:
n=0
cn(xa)n
= c0(xa)0+ c1(xa)1+ c2(xa)2+ ...+ cn(xa)n + ...
= c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NoteIn power series notation:
(xa)0 = 1for all x R (including x = a)
-
Just a note
When we expand:
n=0
cn(xa)n = c0(xa)0+ c1(xa)1+ c2(xa)2+ ...+ cn(xa)n + ...
= c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NoteIn power series notation:
(xa)0 = 1for all x R (including x = a)
-
Just a note
When we expand:
n=0
cn(xa)n = c0(xa)0+ c1(xa)1+ c2(xa)2+ ...+ cn(xa)n + ...
= c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
NoteIn power series notation:
(xa)0 = 1for all x R (including x = a)
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A power series is a function in x, so we can evaluate it at some x = x0 R.
If we plug in x = x0, what do we obtain?We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
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A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
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A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
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A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
-
A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:
n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
-
A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)n
At x = 5:n=0
(3
2
)n
-
A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
-
A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
-
A power series is a function in x, so we can evaluate it at some x = x0 R.If we plug in x = x0, what do we obtain?
We obtain an infinite series.
Example
Evaluaten=0
(x2)n2n
at the following values:
At x = 3:n=0
(1
2
)nAt x = 5:
n=0
(3
2
)n
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Be careful about evaluating a power series in xa at x = a!
Example
Evaluaten=0
(x2)n2n
at x = 2:
n=0
(x2)n2n
= 1
Note
Evaluated at x = a,n=0
cn(xa)n = c0.
-
Be careful about evaluating a power series in xa at x = a!
Example
Evaluaten=0
(x2)n2n
at x = 2:
n=0
(x2)n2n
=
1
Note
Evaluated at x = a,n=0
cn(xa)n = c0.
-
Be careful about evaluating a power series in xa at x = a!
Example
Evaluaten=0
(x2)n2n
at x = 2:
n=0
(x2)n2n
= 1
Note
Evaluated at x = a,n=0
cn(xa)n = c0.
-
Be careful about evaluating a power series in xa at x = a!
Example
Evaluaten=0
(x2)n2n
at x = 2:
n=0
(x2)n2n
= 1
Note
Evaluated at x = a,n=0
cn(xa)n =
c0.
-
Be careful about evaluating a power series in xa at x = a!
Example
Evaluaten=0
(x2)n2n
at x = 2:
n=0
(x2)n2n
= 1
Note
Evaluated at x = a,n=0
cn(xa)n = c0.
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Relevant Question
Given a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
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Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
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Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
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Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n
(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
-
Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
-
Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n
(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
-
Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
-
Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
-
Relevant QuestionGiven a power series, for what values of x = x0 will we obtain a convergentinfinite series?
Example
Givenn=0
(x2)n2n
At x = 3:n=0
(1
2
)n(convergent)
At x = 5:n=0
(3
2
)n(divergent)
At x = 2:n=0
(x2)n2n
= 1 (convergent)
(Solve this example!)
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RemarkA power series in xa is always convergent when x = a.
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Theorem
Letn=0
cn(xa)n be a power series. Exactly one of the following is true:
1 The power series converges only when x = a. (Case 1)2 The power series is absolutely convergent for all x R. (Case 2)3 There exists R > 0 such that the power series is absolutely convergent
for all x such that |xa| R. (Case 3)
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Theorem
Letn=0
cn(xa)n be a power series. Exactly one of the following is true:1 The power series converges only when x = a. (Case 1)2 The power series is absolutely convergent for all x R. (Case 2)3 There exists R > 0 such that the power series is absolutely convergent
for all x such that |xa| R. (Case 3)
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Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.
2 This set of values is called the interval of convergence (IOC) of thepower series.
3 The radius of the interval of convergence is called the radius ofconvergence (ROC) of the power series.
1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
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Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.2 This set of values is called the interval of convergence (IOC) of the
power series.
3 The radius of the interval of convergence is called the radius ofconvergence (ROC) of the power series.
1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
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Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.2 This set of values is called the interval of convergence (IOC) of the
power series.3 The radius of the interval of convergence
is called the radius ofconvergence (ROC) of the power series.
1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
-
Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.2 This set of values is called the interval of convergence (IOC) of the
power series.3 The radius of the interval of convergence is called the radius of
convergence (ROC) of the power series.
1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
-
Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.2 This set of values is called the interval of convergence (IOC) of the
power series.3 The radius of the interval of convergence is called the radius of
convergence (ROC) of the power series.1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
-
Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.2 This set of values is called the interval of convergence (IOC) of the
power series.3 The radius of the interval of convergence is called the radius of
convergence (ROC) of the power series.1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
-
Remark1 The theorem states that the set of all values for which a power series
in xa converges is always an interval centered at a.2 This set of values is called the interval of convergence (IOC) of the
power series.3 The radius of the interval of convergence is called the radius of
convergence (ROC) of the power series.1 Case 1: ROC= 02 Case 2: ROC=3 Case 3: ROC=R
4 For Case 3: The power series may or may not converge at theendpoints of the IOC.
(Illustration for Case 3!)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.
2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.
3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)
= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
To find ROC and IOC
1 We already know x = a is in the interval of convergence.2 Consider the case x 6= a.3 Use ratio or root test. Simplification yields something of the form
limn |xa| f (n)= |xa| limn f (n)
If limn f (n)= 0, then ROC=, IOC= (,). (Case 2)
If limn f (n)=, then ROC= 0, IOC= {a}. (Case 1)
If limn f (n)= L, then solve for ROC. (Case 3)
-
Case 3: If limn f (n)= L, then solve for ROC:
1 ROC: R = 1L
2 IOC:(1) The interval of convergence includes all x such that R < xa
-
Case 3: If limn f (n)= L, then solve for ROC:
1 ROC: R = 1L
2 IOC:(1) The interval of convergence includes all x such that R < xa
-
Case 3: If limn f (n)= L, then solve for ROC:
1 ROC: R = 1L
2 IOC:
(1) The interval of convergence includes all x such that R < xa
-
Case 3: If limn f (n)= L, then solve for ROC:
1 ROC: R = 1L
2 IOC:(1) The interval of convergence includes all x such that R < xa
-
Case 3: If limn f (n)= L, then solve for ROC:
1 ROC: R = 1L
2 IOC:(1) The interval of convergence includes all x such that R < xa
-
Case 3: If limn f (n)= L, then solve for ROC:
1 ROC: R = 1L
2 IOC:(1) The interval of convergence includes all x such that R < xa
-
ExampleFind the radius and interval of convergence of the following:
1n=0
(x2)n2n
2n=0
xn
n!
3n=0
(x+2)n(n+1)3n+1
4n=0
nn(x+1)n
Absolute Convergence, Conditional Convergence and DivergencePower Series