aarjaneiro/planetmath huggingface dataset:
boundedness of terms of power series

boundedness of terms of power series

Theorem

.

If the set

{

a

0

,

a

1

⁢

c

,

a

2

⁢

c

2

,

…

}

of the of a

power series

∑

n

=

0

∞

a

n

⁢

z

n

at the point

z

=

c

is

bounded

(

http://planetmath.org/BoundedInterval

), then the power series

converges

, absolutely (

http://planetmath.org/AbsoluteConvergence

), for any value

z

which

satisfies

|

z

|

<

|

c

|

.

Proof.

By the

assumption

, there exists a positive number

M

such that

|

a

n

⁢

c

n

|

<

M

∀

n

=

0

,

1

,

2

,

…

Thus one gets for the coefficients of the series the estimation

|

a

n

|

<

M

|

c

|

n

.

If now

|

z

|

<

|

c

|

,  one has

|

a

n

⁢

z

n

|

<

M

⁢

|

z

c

|

n

,

and since the

geometric series

∑

n

=

0

∞

|

z

c

|

n

is

convergent

, then also the real series

∑

n

=

0

∞

|

a

n

⁢

z

n

|

converges.

Title

boundedness of terms of power series

Canonical name

BoundednessOfTermsOfPowerSeries

Date of creation

2013-03-22 18:50:44

Last modified on

2013-03-22 18:50:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A30

Classification

msc 30B10

Generated on Fri Feb 9 21:23:33 2018 by

LaTeXML
condition for uniform convergence of sequence of functions

condition for uniform convergence of sequence of functions

Proof of limits of functionsFernando Sanz Gamiz

Theorem 1

.

Let

f

1

,

f

2

,

…

be a

sequence

of real or complex
functions defined on the

interval

[

a

,

b

]

.  The sequence

converges uniformly

to the

limit

function

f

on the
interval

[

a

,

b

]

if and only if

lim

n

→

∞

sup

{

|

f

n

(

x

)

-

f

(

x

)

|

,

a

≤

x

≤

b

}

=

0

.

Proof.

Suppose the sequence converges uniformly. By the very definition of

uniform convergence

, we have that for any

ϵ

there exist

N

such that

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

<

ϵ

2

,

a

≤

x

≤

b

for

⁢

n

>

N

hence

sup

{

|

f

n

(

x

)

-

f

(

x

)

|

,

a

≤

x

≤

b

}

<

ϵ

for

n

>

N

Conversely, suppose the sequence does not converge
uniformly. This means that there is an

ϵ

for which there is
a sequence of

increasing

integers

n

i

,

i

=

1

,

2

,

…

and points

x

n

i

with the corresponding subsequence of functions

f

n

i

such that

|

f

⁢

(

x

n

i

)

-

f

n

i

⁢

(

x

n

i

)

|

>

ϵ

for all

⁢

i

=

1

,

2

,

…

therefore

sup

{

|

f

n

(

x

)

-

f

(

x

)

|

,

a

≤

x

≤

b

}

>

ϵ

for infinitely many

n

.

Consequently, it is not the case that

lim

n

→

∞

sup

{

|

f

n

(

x

)

-

f

(

x

)

|

,

a

≤

x

≤

b

}

=

0

.

∎

Theorem 2

.

The uniform limit of a sequence of

continuous

complex or

real
functions

f

n

in the interval

[

a

,

b

]

is continuous in

[

a

,

b

]

The proof is
here (

http://planetmath.org/LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous

)

Title

condition for uniform convergence of sequence of functions

Canonical name

ConditionForUniformConvergenceOfSequenceOfFunctions

Date of creation

2013-03-22 17:07:49

Last modified on

2013-03-22 17:07:49

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

6

Author

fernsanz (8869)

Entry type

Proof

Classification

msc 40A30

Classification

msc 26A15

Generated on Fri Feb 9 21:23:38 2018 by

LaTeXML
converges uniformly

converges uniformly

Let

X

be a set,

(

Y

,

ρ

)

a metric space and

{

f

n

}

a

sequence

of functions from

X

to

Y

, and

f

:

X

→

Y

another function.

If for every

ε

>

0

there exists an integer

N

such that

ρ

⁢

(

f

n

⁢

(

x

)

,

f

⁢

(

x

)

)

<

ε

for all

x

∈

X

and all

n

>

N

,
then we say that

f

n

converges uniformly

to

f

.

Title

converges uniformly

Canonical name

ConvergesUniformly

Date of creation

2013-03-22 14:01:23

Last modified on

2013-03-22 14:01:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 40A30

Related topic

UniformConvergence

Related topic

AbsoluteConvergence

Generated on Fri Feb 9 21:23:39 2018 by

LaTeXML
derivative of limit function diverges from limit of derivatives

derivative of limit function diverges from limit of derivatives

For a

function

sequence

, one cannot always change the of

http://planetmath.org/node/6209

taking limit and differentiating (

http://planetmath.org/

Differentiate

), i.e. it may well be

lim

n

→

∞

⁡

d

d

⁢

x

⁢

f

n

⁢

(

x

)

≠

d

d

⁢

x

⁢

lim

n

→

∞

⁡

f

n

⁢

(

x

)

,

in the case that a sequence of

continuous

(and

differentiable

) functions

converges uniformly

; cf.

Theorem

2 of the parent entry (

http://planetmath.org/LimitFunctionOfSequence

).

Example.

The function sequence

f

n

(

x

)

:=

∑

j

=

1

n

x

3

(

1

+

x

2

)

j

=

x

-

x

(

1

+

x

2

)

n

(

n

=

1

,

2

,

3

,

…

)

(1)

provides an

instance

; we consider it on the

interval

[

-

1

,

1

]

.  It’s a question of

partial sum

the converging

geometric series

x

3

1

+

x

2

+

x

3

(

1

+

x

2

)

2

+

x

3

(

1

+

x

2

)

2

+

…

(although one cannot use Weierstrass’ criterion of

uniform convergence

).  Since
the limit function is

f

⁢

(

x

)

:=

lim

n

→

∞

⁡

(

x

-

x

(

1

+

x

2

)

n

)

=

x

∀

x

∈

[

-

1

,

1

]

,

we have

sup

[

-

1

,

1

]

⁡

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

=

sup

[

-

1

,

1

]

⁡

|

x

|

(

1

+

x

2

)

n

⟶

0

as

⁢

n

→

∞

,

which means by Theorem 1 of the parent entry (

http://planetmath.org/LimitFunction

) that the sequence (1) converges uniformly on the interval to the

identity function

.  Further, the members of the sequence are continuous and differentiable.  Furthermore,

f

n

′

⁢

(

x

)

=

1

-

1

+

(

1

-

2

⁢

n

)

⁢

x

2

(

1

+

x

2

)

n

+

1

,

whence

lim

n

→

∞

f

n

′

(

x

)

=

1

(

x

≠

0

)

.

But in the point

x

=

0

we have

lim

n

→

∞

⁡

f

n

′

⁢

(

0

)

=

lim

n

→

∞

⁡

0

=

0

,

which says that the limit of

derivative

sequence of (1) is

discontinuous

in the origin.  Because

f

′

⁢

(

x

)

≡

1

,

we may write

lim

n

→

∞

⁡

d

d

⁢

x

⁢

f

n

≠

d

d

⁢

x

⁢

lim

n

→

∞

⁡

f

n

.

Title

derivative of limit function diverges from limit of derivatives

Canonical name

DerivativeOfLimitFunctionDivergesFromLimitOfDerivatives

Date of creation

2013-03-22 19:00:29

Last modified on

2013-03-22 19:00:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Example

Classification

msc 40A30

Classification

msc 26A15

Synonym

limit of derivatives diverges from derivative of limit function

Related topic

GrowthOfExponentialFunction

Generated on Fri Feb 9 21:23:42 2018 by

LaTeXML
indirect proof of identity theorem of power series

indirect proof of identity theorem of power series

∑

n

=

0

∞

a

n

⁢

(

z

-

z

0

)

n

=

∑

n

=

0

∞

b

n

⁢

(

z

-

z

0

)

n

(1)

is valid in the set of points

z

presumed in the theorem (

http://

planetmath

.org/IdentityTheoremOfPowerSeries

) to be proved.

Antithesis:  There are integers

n

such that

a

n

≠

b

n

;  let

ν

(

≧

0

) be least of them.

We can choose from the point set an infinite

sequence

z

1

,

z

2

,

z

3

,

…

which

converges

to

z

0

with

z

n

≠

z

0

for every

n

.  Let

z

in the equation (1) belong to

{

z

1

,

z

2

,

z

3

,

…

}

and let’s divide both of (1) by

(

z

-

z

0

)

ν

which is distinct from zero; we then have

a

ν

+

a

ν

+

1

⁢

(

z

-

z

0

)

+

a

ν

+

2

⁢

(

z

-

z

0

)

2

+

…

⏟

f

⁢

(

z

)

=

b

ν

+

b

ν

+

1

⁢

(

z

-

z

0

)

+

b

ν

+

2

⁢

(

z

-

z

0

)

2

+

…

⏟

g

⁢

(

z

)

(2)

Let here

z

to tend

z

0

along the points

z

1

,

z

2

,

z

3

,

…

, i.e. we take the limits

lim

n

→

∞

⁡

f

⁢

(

z

n

)

and

lim

n

→

∞

⁡

g

⁢

(

z

n

)

.  Because the

sum of power

series is always a

continuous function

, we see that in (2),

left

⁢

side

⟶

f

⁢

(

z

0

)

=

a

ν

and

right

⁢

side

⟶

g

⁢

(

z

0

)

=

b

ν

But all the time, the left and of (2) are equal, and thus also the limits.  So we must have

a

ν

=

b

ν

,  contrary to the antithesis.  We conclude that the antithesis is wrong.  This settles the proof.

Note.

I learned this proof from my venerable teacher, the number-theorist

Kustaa Inkeri

(1908–1997).

Title

indirect proof of identity theorem of power series

Canonical name

IndirectProofOfIdentityTheoremOfPowerSeries

Date of creation

2013-03-22 16:47:48

Last modified on

2013-03-22 16:47:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Proof

Classification

msc 40A30

Classification

msc 30B10

Generated on Fri Feb 9 21:23:44 2018 by

LaTeXML
limit laws for uniform convergence

limit laws for uniform convergence

As might be expected, the usual (

pointwise

) limit laws — for instance, that the sum of limits is the limit of sums — have analogues for uniform limits. The laws for uniform limits are usually not mentioned in elementary textbooks, but they are useful in situations where having

uniform convergence

is crucial,
such as when working with infinite sums or

products

of

holomorphic

or

meromorphic functions

.

The uniform laws may be derived simply by reinterpreting the pointwise proofs from any

calculus

text, with some extra conditions. Some of these results are listed below.

In the following,

X

is a

metric space

, and

Y

is another metric space (with the appropriate operations defined), while

f

n

:

X

→

Y

and

g

n

:

X

→

Y

(with

n

∈

ℕ

)
are

functions

that

converge uniformly

on

X

to

f

:

X

→

Y

and

g

:

X

→

Y

, respectively. (For example,

X

and

Y

may be

ℂ

, the

complex plane

.)

Theorem 1

.

If

Y

is a

normed vector space

,
then

f

n

+

g

n

uniformly

converge

to

f

+

g

.

Theorem 2

.

If

Y

is a

Banach algebra

, and both

f

⁢

(

X

)

and

g

⁢

(

X

)

are

bounded

,
then

f

n

⋅

g

n

uniformly converge to

f

⋅

g

.

Theorem 3

.

Let

Z

be another metric space.
Suppose that

X

is

compact

,

Y

is

locally compact

,
and

f

:

X

→

Y

, and

h

:

Y

→

Z

are

continuous functions

. Then

h

∘

f

n

converge to

h

∘

f

uniformly.

Title

limit laws for uniform convergence

Canonical name

LimitLawsForUniformConvergence

Date of creation

2013-03-22 15:23:06

Last modified on

2013-03-22 15:23:06

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 40A30

Generated on Fri Feb 9 21:23:46 2018 by

LaTeXML
Loewner ordering

Loewner ordering

Let

H

be a

Hilbert space

, and let

X

,

Y

∈

Sym

⁢

(

E

)

be

symmetric operators

on

H

.

We define the Loewner order

<

F

on

Sym

⁢

(

E

)

by declaring

X

<

F

Y

if

X

-

Y

is a

positive semidefinite

invertible

bounded operator

on

H

, and

X

<

F

Y

if

X

-

Y

is a positive semidefinite invertible bounded operator on

H

.

Title

Loewner ordering

Canonical name

LoewnerOrdering

Date of creation

2013-03-22 14:40:15

Last modified on

2013-03-22 14:40:15

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 40A30

Defines

Loewner order

Generated on Fri Feb 9 21:23:47 2018 by

LaTeXML
Löwner partial ordering

Löwner partial ordering

Let

A

and

B

be two

Hermitian matrices

of the same size. If

A

-

B

is

positive semidefinite

we write

A

≥

B

,

or

B

≤

A

.

Then

≥

is a

partial ordering

,
referred to as

Löwner partial ordering

,
on the set of Hermitian matrices.

Title

Löwner partial ordering

Canonical name

LownerPartialOrdering

Date of creation

2013-03-22 13:35:03

Last modified on

2013-03-22 13:35:03

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 40A30

Related topic

MatrixMonotone

Generated on Fri Feb 9 21:23:49 2018 by

LaTeXML
Löwner’s theorem

Löwner’s theorem

A real function

f

on an

interval

I

is

matrix monotone

if and only if it is

real
analytic

and has (

complex

)

analytic continuations

to the upper
and lower half planes such that

ℑ

⁡

(

f

)

>

0

in the

upper half plane

.

(Löwner 1934)

Title

Löwner’s theorem

Canonical name

LownersTheorem

Date of creation

2013-03-22 13:34:49

Last modified on

2013-03-22 13:34:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 40A30

Generated on Fri Feb 9 21:23:50 2018 by

LaTeXML
matrix monotone

matrix monotone

A

real function

f

on a real interval

I

is said to be

matrix monotone

of

n

, if

A

≤

L

B

⇒

f

⁢

(

A

)

≤

f

⁢

(

B

)

(1)

for all Hermitian

n

×

n

matrices

A

,

B

with spectra contained in

I

. Here

≤

L

denotes the

Loewner order

, and the notation

f

⁢

(

A

)

is explained in the entry

functional calculus for Hermitian matrices

.

Title

matrix monotone

Canonical name

MatrixMonotone

Date of creation

2013-03-22 13:34:57

Last modified on

2013-03-22 13:34:57

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 40A30

Related topic

ALeqBForHermitianMatricesAB

Related topic

OperatorMonotone

Generated on Fri Feb 9 21:23:52 2018 by

LaTeXML
one-sided continuity by series

one-sided continuity by series

Theorem.

If the function series

∑

n

=

1

∞

f

n

⁢

(

x

)

(1)

is

uniformly convergent

on the

interval

[

a

,

b

]

,  on which the

f

n

⁢

(

x

)

are

continuous

from the right or from the left, then the sum function

S

⁢

(

x

)

of the series has the same property.

Proof.

Suppose that the terms

f

n

⁢

(

x

)

are continuous from the right.  Let

ε

be any positive number and

S

⁢

(

x

)

:=

S

n

⁢

(

x

)

+

R

n

+

1

⁢

(

x

)

,

where

S

n

⁢

(

x

)

is the

n

th

partial sum

of (1) (

n

=

1

,

2

,

…

).  The

uniform convergence

implies the existence of a number

n

ε

such that on the whole interval we have

|

R

n

+

1

⁢

(

x

)

|

<

ε

3

when

⁢

n

>

n

ε

.

Let now

n

>

n

ε

and

x

0

,

x

0

+

h

∈

[

a

,

b

]

with

h

>

0

.  Since every

f

n

⁢

(

x

)

is continuous from the right in

x

0

, the same is true for the finite sum

S

n

⁢

(

x

)

, and therefore there exists a number

δ

ε

such that

|

S

n

⁢

(

x

0

+

h

)

-

S

n

⁢

(

x

0

)

|

<

ε

3

when

⁢

0

<

h

<

δ

ε

.

Thus we obtain that

|

S

⁢

(

x

0

+

h

)

-

S

⁢

(

x

0

)

|

=

|

[

S

n

(

x

0

+

h

)

-

S

n

(

x

0

)

]

+

R

n

+

1

(

x

0

+

h

)

-

R

n

+

1

(

x

0

|

≦

|

S

n

⁢

(

x

0

+

h

)

-

S

n

⁢

(

x

0

)

|

+

|

R

n

+

1

⁢

(

x

0

+

h

)

|

+

|

R

n

+

1

⁢

(

x

0

)

|

<

ε

3

+

ε

3

+

ε

3

=

ε

as soon as

0

<

h

<

δ

ε

.

This means that

S

is continuous from the right in an arbitrary point

x

0

of

[

a

,

b

]

.

Analogously, one can prove the assertion concerning the

continuity

from the left.

Title

one-sided continuity by series

Canonical name

OnesidedContinuityBySeries

Date of creation

2013-03-22 18:34:03

Last modified on

2013-03-22 18:34:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A30

Classification

msc 26A03

Synonym

one-sided continuity of series with terms one-sidedly continuous

Related topic

OneSidedContinuity

Related topic

SumFunctionOfSeries

Generated on Fri Feb 9 21:23:55 2018 by

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operator monotone

operator monotone

A function is said to be

operator monotone

if it is

matrix monotone

of every order.

Title

operator monotone

Canonical name

OperatorMonotone

Date of creation

2013-03-22 13:35:00

Last modified on

2013-03-22 13:35:00

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 40A30

Related topic

MatrixMonotone

Generated on Fri Feb 9 21:23:56 2018 by

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point preventing uniform convergence

point preventing uniform convergence

Theorem.

If the

sequence

f

1

,

f

2

,

f

3

,

…

of real functions

converges

at each point of the

interval

[

a

,

b

]

but does not

converge uniformly

on this interval, then there exists at least one point

x

0

of the interval such that the function sequence converges uniformly on no closed sub-interval of

[

a

,

b

]

containing

x

0

.

Proof.

Let the limit function of the sequence on the interval

[

a

,

b

]

be

f

. According the entry

uniform convergence on union interval

, the sequence can not converge uniformly to

f

both

half-intervals

[

a

,

a

+

b

2

]

and

[

a

+

b

2

,

b

]

,  since otherwise it would do it on the union

[

a

,

b

]

. Denote by

[

a

1

,

b

1

]

the first (fom left) of those half-intervals on which the

convergence

is not uniform. We have

[

a

,

b

]

⊃

[

a

1

,

b

1

]

. Then the interval

[

a

1

,

b

1

]

is halved and chosen its half-interval

[

a

2

,

b

2

]

on which the convergence is not uniform. We can continue similarly arbitrarily far and obtain a unique endless sequence

[

a

,

b

]

⊃

[

a

1

,

b

1

]

⊃

[

a

2

,

b

2

]

⊃

…

of nested intervals on which the convergence of the function sequence is not uniform, and besides the length of the intervals tend to zero:

lim

n

→

∞

⁡

(

b

n

-

a

n

)

=

lim

n

→

∞

⁡

b

-

a

2

n

=

0

.

The

nested interval theorem

thus gives a unique

real number

x

0

belonging to each of the intervals

[

a

,

b

]

and

[

a

n

,

b

n

]

. Then

lim

n

→

∞

⁡

a

n

=

x

0

=

lim

n

→

∞

⁡

b

n

.  Let us choose

α

and

β

such that

a

≦

α

≦

x

0

≦

β

≦

b

. There exist the integers

n

1

and

n

2

such that

|

a

n

-

x

0

|

=

x

0

-

a

n

≦

x

0

-

α

when

⁢

n

>

n

1

|

b

n

-

x

0

|

=

b

n

-

x

0

≦

β

-

x

0

when

⁢

n

>

n

2

.

Therefore

α

≦

a

n

≦

x

0

≦

b

n

≦

β

when

⁢

n

>

max

⁡

{

n

1

,

n

2

}

.

This means that

f

n

→

f

not uniformly on

[

a

n

,

b

n

]

⊂

[

α

,

β

]

, whence the function sequence does not converge uniformly on the arbitrarily chosen

subinterval

[

α

,

β

]

of

[

a

,

b

]

containing

x

0

.

Title

point preventing uniform convergence

Canonical name

PointPreventingUniformConvergence

Date of creation

2013-03-22 17:27:18

Last modified on

2013-03-22 17:27:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A30

Related topic

NotUniformlyContinuousFunction

Related topic

LimitFunctionOfSequence

Generated on Fri Feb 9 21:23:59 2018 by

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pointwise convergence

pointwise convergence

Let

X

be any set, and let

Y

be a

topological space

.
A

sequence

f

1

,

f

2

,

…

of functions mapping

X

to

Y

is said to be

pointwise

convergent

(or simply convergent) to another function

f

, if the sequence

f

n

⁢

(

x

)

converges

to

f

⁢

(

x

)

for each

x

in

X

. This is usually denoted by

f

n

→

f

.

Title

pointwise convergence

Canonical name

PointwiseConvergence

Date of creation

2013-03-22 13:15:28

Last modified on

2013-03-22 13:15:28

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Definition

Classification

msc 40A30

Defines

pointwise

Generated on Fri Feb 9 21:24:01 2018 by

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power series

power series

A

power series

is a series of the form

∑

k

=

0

∞

a

k

⁢

(

x

-

x

0

)

k

,

with

a

k

,

x

0

∈

ℝ

or

∈

ℂ

. The

a

k

are called the

coefficients

and

x

0

the center of the power series.

a

0

is called the

constant term

.

Where it

converges

the power series defines a

function

, which can thus be represented by a power series. This is what power series are usually used for.
Every power series is

convergent

at least at

x

=

x

0

where it converges to

a

0

. In

addition

it is absolutely and

uniformly convergent

in the region

{

x

∣

|

x

-

x

0

|

<

r

}

, with

r

=

lim inf

k

→

∞

⁡

1

|

a

k

|

k

It is divergent for every

x

with

|

x

-

x

0

|

>

r

. For

|

x

-

x

0

|

=

r

no general predictions can be made. If

r

=

∞

, the power series converges absolutely and uniformly for every real or

complex

x

.

The

real number

r

is called the

radius of convergence

of the power series.

Examples of power series are:

•

Taylor series

, for example:

e

x

=

∑

k

=

0

∞

x

k

k

!

.

•

The

geometric series

:

1

1

-

x

=

∑

k

=

0

∞

x

k

,

with

|

x

|

<

1

.

Power series have some important :

•

If a power series converges for a

z

0

∈

ℂ

then it also converges for all

z

∈

ℂ

with

|

z

-

x

0

|

<

|

z

0

-

x

0

|

.

•

Also, if a power series diverges for some

z

0

∈

ℂ

then it diverges for all

z

∈

ℂ

with

|

z

-

x

0

|

>

|

z

0

-

x

0

|

.

•

For

|

x

-

x

0

|

<

r

Power series can be added by adding coefficients and multiplied in the obvious way:

∑

k

=

0

∞

a

k

⁢

(

x

-

x

o

)

k

⋅

∑

l

=

0

∞

b

j

⁢

(

x

-

x

0

)

j

=

a

0

⁢

b

0

+

(

a

0

⁢

b

1

+

a

1

⁢

b

0

)

⁢

(

x

-

x

0

)

+

(

a

0

⁢

b

2

+

a

1

⁢

b

1

+

a

2

⁢

b

0

)

⁢

(

x

-

x

0

)

2

⁢

…

.

•

(Uniqueness) If two power series are equal and their are the same, then their coefficients must be equal.

•

Power series can be termwise differentiated and integrated. These

operations

keep the radius of convergence.

Title

power series

Canonical name

PowerSeries

Date of creation

2013-03-22 12:32:55

Last modified on

2013-03-22 12:32:55

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

23

Author

azdbacks4234 (14155)

Entry type

Definition

Classification

msc 40A30

Classification

msc 30B10

Related topic

TaylorSeries

Related topic

FormalPowerSeries

Related topic

TermwiseDifferentiation

Related topic

AbelsLimitTheorem

Defines

constant term

Generated on Fri Feb 9 21:24:04 2018 by

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