MATH 38Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

Outline

1 Absolute Convergence, Conditional Convergence and Divergence

2 Power Series

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!!!

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!!!

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!!!

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!!!

ExampleDetermine whether the given series is absolutely convergent, conditionallyconvergent or divergent.

1n=1

(1)n(2n+1)n2

2n=1

(1)n3n+1nn

Outline

1 Absolute Convergence, Conditional Convergence and Divergence

2 Power Series

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.)

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.)

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+ 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+ ...

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+ ...

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)

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)

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

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

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.

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!)

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!)

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!)

RemarkA power series in xa is always convergent when x = a.

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)

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)

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!)

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 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