aarjaneiro/planetmath huggingface dataset:
proof of Abel’s limit theorem

proof of Abel’s limit theorem

Without loss of generality we may assume

r

=

1

, because otherwise we can set

a

n

′

:=

a

r

n

, so that

∑

a

n

′

⁢

x

n

has radius

1

and

∑

a

′

is

convergent

if and only if

∑

a

n

⁢

r

n

is.
We now have to show that the function

f

⁢

(

x

)

generated by

∑

a

n

⁢

x

n

(with

r

=

1

)is

continuous

from below at

x

=

1

if it is defined there.
Let

s

:=

∑

a

n

. We have to show that

lim

x

→

1

-

⁡

f

⁢

(

x

)

=

s

.

If

|

x

|

<

1

we have:

s

-

f

⁢

(

x

)

=

∑

n

=

0

∞

a

n

-

∑

n

=

0

∞

a

n

⁢

x

n

=

∑

n

=

0

∞

(

1

-

x

n

)

⁢

a

n

=

(

1

-

x

)

⁢

∑

n

=

1

∞

(

x

n

-

1

+

x

n

-

2

+

…

+

x

+

1

)

⁢

a

n

=

(

1

-

x

)

⁢

∑

n

=

0

∞

(

s

-

s

n

)

⁢

x

n

with

s

n

:=

∑

i

=

0

n

a

i

. Now, since

s

-

s

n

→

0

as

n

→

∞

we can choose an

N

for every

ε

>

0

such that

|

s

-

s

n

|

<

ε

2

for all

m

>

N

. So for every

0

<

x

<

1

we have:

|

s

-

f

⁢

(

x

)

|

<

(

1

-

x

)

⁢

∑

n

=

0

m

|

r

n

|

⁢

x

n

+

ε

2

⁢

(

1

-

x

)

⁢

∑

n

=

m

+

1

∞

x

n

<

(

1

-

x

)

⁢

∑

n

=

0

m

|

r

n

|

+

ε

2

.

This is smaller than

ε

for all

x

<

1

sufficiently close to

1

, which proves

lim

x

→

r

-

⁡

∑

a

n

⁢

x

n

=

∑

a

n

⁢

r

n

=

∑

lim

x

→

r

-

⁡

a

n

⁢

x

n

.

Title

proof of Abel’s limit theorem

Canonical name

ProofOfAbelsLimitTheorem

Date of creation

2013-03-22 14:09:40

Last modified on

2013-03-22 14:09:40

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Proof

Classification

msc 40A30

Related topic

ProofOfAbelsConvergenceTheorem

Generated on Fri Feb 9 21:24:07 2018 by

LaTeXML
proof of composition limit law for uniform convergence

proof of composition limit law for uniform convergence

Theorem 1

.

Let

X

,

Y

,

Z

be

metric spaces

, with

X

compact

and

Y

locally compact

.
If

f

n

:

X

→

Y

is a

sequence

of

functions

converging uniformly
to a

continuous function

f

:

X

→

Y

, and

h

:

Y

→

Z

is

continuous

, then

h

∘

f

n

converge

to

h

∘

f

uniformly.

Proof.

Let

K

denote the compact set

f

⁢

(

X

)

⊆

Y

. By local compactness of

Y

,for each point

y

∈

K

, there is an open

neighbourhood

U

y

of

y

such that

U

y

¯

is compact. The neighbourhoods

U

y

cover

K

, so there is a finite

subcover

U

y

1

,

…

,

U

y

n

covering

K

. Let

U

=

⋃

i

U

y

i

⊇

K

. Evidently

U

¯

=

⋃

i

U

y

i

¯

is compact.

Next, let

V

be the

δ

0

-neighbourhood of

K

contained

in

U

, for some

δ

0

>

0

.

V

¯

is compact, since it is contained in

U

¯

.

Now let

ϵ

>

0

be given.

h

is

uniformly continuous

on

V

¯

, so
there exists a

δ

>

0

such that
when

y

,

y

′

∈

V

¯

and

d

⁢

(

y

,

y

′

)

<

δ

,
we have

d

⁢

(

g

⁢

(

y

)

,

g

⁢

(

y

′

)

)

<

ϵ

.

From the

uniform convergence

of

f

n

, choose

N

so that
when

n

≥

N

,

d

⁢

(

f

n

⁢

(

x

)

,

f

⁢

(

x

)

)

<

min

⁡

(

δ

,

δ

0

)

for all

x

∈

X

.
Since

f

⁢

(

x

)

∈

K

, it follows that

f

n

⁢

(

x

)

is inside the

δ

0

-neighbourhood of

K

, i.e. both

y

=

f

n

⁢

(

x

)

and

y

′

=

f

⁢

(

x

)

are both in

V

. Thus

d

⁢

(

g

⁢

(

f

n

⁢

(

x

)

)

,

g

⁢

(

f

⁢

(

x

)

)

)

<

ϵ

when

n

≥

N

,
uniformly for all

x

∈

X

.
∎

Title

proof of composition limit law for uniform convergence

Canonical name

ProofOfCompositionLimitLawForUniformConvergence

Date of creation

2013-03-22 15:23:08

Last modified on

2013-03-22 15:23:08

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Proof

Classification

msc 40A30

Generated on Fri Feb 9 21:24:10 2018 by

LaTeXML
proof of identity theorem of power series

proof of identity theorem of power series

We can prove the

identity theorem

for

power series

using

divided differences

. From amongst the points
at which the two series are equal, pick a

sequence

{

w

k

}

k

=

0

∞

which satisfies the following
three conditions:

1.

lim

k

→

∞

⁡

w

k

=

z

0

2.

w

m

=

w

n

if and only if

m

=

n

.

3.

w

k

≠

z

0

for all

k

.

Let

f

be the

function

determined by one power series
and let

g

be the function determined by the other
power series:

f

⁢

(

z

)

=

∑

n

=

0

∞

a

n

⁢

(

z

-

z

0

)

n

g

⁢

(

z

)

=

∑

n

=

0

∞

b

n

⁢

(

z

-

z

0

)

n

Because formation of divided differences involves
finite sums and dividing by differences of

w

k

’s
(which all differ from zero by condition 2 above,
so it is legitimate to divide by them), we may
carry out the formation of finite diffferences
on a term-by-term basis. Using the result about

divided differences of powers

, we have

Δ

m

⁢

f

⁢

[

w

k

,

…

,

w

k

+

m

]

=

∑

n

=

m

∞

a

n

⁢

D

m

⁢

n

⁢

k

Δ

m

⁢

f

⁢

[

w

k

,

…

,

w

k

+

m

]

=

∑

n

=

m

∞

b

n

⁢

D

m

⁢

n

⁢

k

where

D

m

⁢

n

⁢

k

=

∑

j

0

+

…

⁢

j

m

=

n

-

m

(

w

k

-

z

0

)

j

0

⁢

⋯

⁢

(

w

k

+

m

-

z

0

)

j

m

.

Note that

lim

k

→

i

⁢

n

⁢

f

⁢

t

⁢

y

⁡

D

m

⁢

n

⁢

k

=

0

when

m

>

n

, but

D

m

⁢

m

⁢

k

=

1

. Since
power series

converge uniformly

, we may
intechange limit and summation to conclude

lim

k

→

∞

⁡

Δ

m

⁢

f

⁢

[

w

k

,

…

,

w

k

+

m

]

=

∑

n

=

m

∞

a

n

⁢

lim

k

→

∞

⁡

D

m

⁢

n

⁢

k

=

a

m

lim

k

→

∞

⁡

Δ

m

⁢

g

⁢

[

w

k

,

…

,

w

k

+

m

]

=

∑

n

=

m

∞

b

n

⁢

lim

k

→

∞

⁡

D

m

⁢

n

⁢

k

=

b

m

.

Since, by design,

f

⁢

(

w

k

)

=

g

⁢

(

w

k

)

, we have

Δ

m

⁢

f

⁢

[

w

k

,

…

,

w

k

+

m

]

=

Δ

m

⁢

g

⁢

[

w

k

,

…

,

w

k

+

m

]

,

hence

a

m

=

b

m

for all

m

.

Title

proof of identity theorem of power series

Canonical name

ProofOfIdentityTheoremOfPowerSeries1

Date of creation

2013-03-22 16:48:46

Last modified on

2013-03-22 16:48:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 40A30

Classification

msc 30B10

Generated on Fri Feb 9 21:24:13 2018 by

LaTeXML
proof of radius of convergence

proof of radius of convergence

According to Cauchy’s

root test

a

power series

is

absolutely convergent

if

lim sup

k

→

∞

⁡

|

a

k

⁢

(

x

-

x

0

)

k

|

k

=

|

x

-

x

0

|

⁢

lim sup

k

→

∞

⁡

|

a

k

|

k

<

1

.

This is obviously true if

|

x

-

x

0

|

<

1

lim sup

k

→

∞

⁡

|

a

k

|

k

=

lim inf

k

→

∞

⁡

1

|

a

k

|

k

=

.

In the same way we see that the series is divergent if

|

x

-

x

0

|

>

lim inf

k

→

∞

⁡

1

|

a

k

|

k

,

which means that the right hand side is the

radius of convergence

of the power series.
Now from the

ratio test

we see that the power series is absolutely convergent if

lim

k

→

∞

⁡

|

a

k

+

1

⁢

(

x

-

x

0

)

k

+

1

a

k

⁢

(

x

-

x

0

)

k

|

=

|

x

-

x

0

|

⁢

lim

k

→

∞

⁡

|

a

k

+

1

a

k

|

<

1

.

Again this is true if

|

x

-

x

0

|

<

lim

k

→

∞

⁡

|

a

k

a

k

+

1

|

.

The series is divergent if

|

x

-

x

0

|

>

lim

k

→

∞

⁡

|

a

k

a

k

+

1

|

,

as follows from the ratio test in the same way. So we see that in this way too we can the radius of convergence.

Title

proof of radius of convergence

Canonical name

ProofOfRadiusOfConvergence

Date of creation

2013-03-22 13:21:50

Last modified on

2013-03-22 13:21:50

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Proof

Classification

msc 40A30

Classification

msc 30B10

Generated on Fri Feb 9 21:24:16 2018 by

LaTeXML
proof of Weierstrass’ criterion of uniform convergence

proof of Weierstrass’ criterion of uniform convergence

The assumption that

|

f

n

⁢

(

x

)

|

≤

M

n

for every

x

guarantees that each numerical series

∑

n

f

n

⁢

(

x

)

converges absolutely

. We call the limit

f

⁢

(

x

)

.

To see that the

convergence

is uniform: let

ϵ

>

0

. Then there exists

K

such that

n

>

K

implies

∑

n

>

K

M

n

<

ϵ

. Now, if

k

>

K

,

|

f

⁢

(

x

)

-

∑

n

=

1

k

f

n

⁢

(

x

)

|

=

|

∑

n

>

k

f

n

⁢

(

x

)

|

≤

∑

n

>

k

|

f

n

⁢

(

x

)

|

≤

∑

n

>

k

M

n

<

ϵ

The

ϵ

does not depend on

x

, so the convergence is uniform.

Title

proof of Weierstrass’ criterion of

uniform convergence

Canonical name

ProofOfWeierstrassCriterionOfUniformConvergence

Date of creation

2013-03-22 16:26:28

Last modified on

2013-03-22 16:26:28

Owner

argerami (15454)

Last modified by

argerami (15454)

Numerical id

4

Author

argerami (15454)

Entry type

Proof

Classification

msc 40A30

Classification

msc 26A15

Generated on Fri Feb 9 21:24:18 2018 by

LaTeXML
radius of convergence

radius of convergence

To the

power series

∑

k

=

0

∞

a

k

⁢

(

x

-

x

0

)

k

(1)

there exists a number

r

∈

[

0

,

∞

]

, its

radius of convergence

, such that the series

converges absolutely

for all (real or

complex

) numbers

x

with

|

x

-

x

0

|

<

r

and

diverges

whenever

|

x

-

x

0

|

>

r

. This is known as

Abel’s theorem on power series

. (For

|

x

-

x

0

|

=

r

no general statements can be made.)

The radius of convergence is given by:

r

=

lim inf

k

→

∞

⁡

1

|

a

k

|

k

(2)

and can also be computed as

r

=

lim

k

→

∞

⁡

|

a

k

a

k

+

1

|

,

(3)

if this limit exists.

It follows from the Weierstrass

M

-test (

http://planetmath.org/WeierstrassMTest

) that for any radius

r

′

smaller than the radius of convergence, the power series

converges uniformly

within the closed disk of radius

r

′

.

Title

radius of convergence

Canonical name

RadiusOfConvergence

Date of creation

2013-03-22 12:32:59

Last modified on

2013-03-22 12:32:59

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

13

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 40A30

Classification

msc 30B10

Synonym

Abel’s theorem on power series

Related topic

ExampleOfAnalyticContinuation

Related topic

NielsHenrikAbel

Generated on Fri Feb 9 21:24:20 2018 by

LaTeXML
the limit of a uniformly convergent sequence of continuous functions is continuous

the limit of a uniformly convergent sequence of continuous functions is continuous

Theorem

.

The limit of a uniformly convergent

sequence

of

continuous functions

is

continuous

.

Proof.

Let

f

n

,

f

:

X

→

Y

, where

(

X

,

ρ

)

and

(

Y

,

d

)

are

metric spaces

. Suppose

f

n

→

f

uniformly and each

f

n

is continuous. Then given any

ϵ

>

0

, there exists

N

such that

n

>

N

implies

d

⁢

(

f

⁢

(

x

)

,

f

n

⁢

(

x

)

)

<

ϵ

3

for all

x

. Pick an arbitrary

n

larger than

N

. Since

f

n

is continuous, given any point

x

0

, there exists

δ

>

0

such that

0

<

ρ

⁢

(

x

,

x

0

)

<

δ

implies

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

x

0

)

)

<

ϵ

3

. Therefore, given any

x

0

and

ϵ

>

0

, there exists

δ

>

0

such that

0

<

ρ

⁢

(

x

,

x

0

)

<

δ

⇒

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

x

0

)

)

≤

d

⁢

(

f

⁢

(

x

)

,

f

n

⁢

(

x

)

)

+

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

x

0

)

)

+

d

⁢

(

f

n

⁢

(

x

0

)

,

f

⁢

(

x

0

)

)

<

ϵ

.

Therefore,

f

is continuous.

The theorem also generalizes to when

X

is an arbitrary

topological space

. To generalize it to

X

an arbitrary topological space, note that if

d

⁢

(

f

n

⁢

(

x

)

,

f

⁢

(

x

)

)

<

ϵ

/

3

for all

x

, then

x

0

∈

f

n

-

1

⁢

(

B

ϵ

/

3

⁢

(

f

n

⁢

(

x

0

)

)

)

⊆

f

-

1

⁢

(

B

ϵ

⁢

(

f

⁢

(

x

0

)

)

)

,
so

f

-

1

⁢

(

B

ϵ

⁢

(

f

⁢

(

x

0

)

)

)

is a

neighbourhood

of

x

0

. Here

B

ϵ

⁢

(

y

)

denote the open ball of radius

ϵ

, centered at

y

.

Title

the limit of a uniformly convergent sequence of continuous functions is continuous

Canonical name

TheLimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous

Date of creation

2013-03-22 15:21:58

Last modified on

2013-03-22 15:21:58

Owner

neapol1s (9480)

Last modified by

neapol1s (9480)

Numerical id

13

Author

neapol1s (9480)

Entry type

Theorem

Classification

msc 40A30

Related topic

LimitFunctionOfSequence

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

LaTeXML
uniform convergence

uniform convergence

Let

X

be any set, and let

(

Y

,

d

)

be a metric space.
A

sequence

f

1

,

f

2

,

…

of functions mapping

X

to

Y

is said to be

uniformly convergent

to another function

f

if, for each

ε

>

0

, there exists

N

such that, for all

x

and all

n

>

N

, we have

d

⁢

(

f

n

⁢

(

x

)

,

f

⁢

(

x

)

)

<

ε

.
This is denoted by

f

n

→

𝑢

f

, or “

f

n

→

f

uniformly” or, less frequently, by

f

n

⇉

f

.

Title

uniform convergence

Canonical

name

UniformConvergence

Date of creation

2013-03-22 13:13:49

Last modified on

2013-03-22 13:13:49

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

14

Author

Koro (127)

Entry type

Definition

Classification

msc 40A30

Related topic

CompactOpenTopology

Related topic

ConvergesUniformly

Defines

uniformly convergent

Generated on Fri Feb 9 21:24:26 2018 by

LaTeXML
uniform convergence on union interval

uniform convergence on union interval

Theorem.

If

a

<

b

<

c

and the

sequence

f

1

,

f

2

,

f

3

,

…

of

real functions

converges uniformly

both on the interval

[

a

,

b

]

and on the interval

[

b

,

c

]

,  then the

function

sequence converges uniformly also on the union (

http://planetmath.org/Union

) interval

[

a

,

c

]

.

Proof.

We have the limit functions

f

a

⁢

b

:=

lim

n

→

∞

⁡

f

n

on

[

a

,

b

]

and

f

b

⁢

c

:=

lim

n

→

∞

⁡

f

n

. It follows that

f

a

⁢

b

⁢

(

b

)

=

lim

n

→

∞

⁡

f

n

⁢

(

b

)

=

f

b

⁢

c

⁢

(

b

)

.

Define the new function

f

⁢

(

x

)

:=

{

f

a

⁢

b

⁢

(

x

)

∀

x

∈

[

a

,

b

]

,

f

b

⁢

c

⁢

(

x

)

∀

x

∈

[

b

,

c

]

.

Choose an arbitrary positive number

ε

. The supposed

uniform convergences

on the intervals

[

a

,

b

]

and

[

b

,

c

]

imply the existence of the numbers

n

1

⁢

(

ε

)

and

n

2

⁢

(

ε

)

such that

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

<

ε

⁢

∀

x

∈

[

a

,

b

]

,

when

⁢

n

>

n

1

⁢

(

ε

)

and

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

<

ε

⁢

∀

x

∈

[

b

,

c

]

,

when

⁢

n

>

n

2

⁢

(

ε

)

.

If one takes

n

>

max

⁡

{

n

1

⁢

(

ε

)

,

n

2

⁢

(

ε

)

}

,  then one has simultaneously on both intervals

[

a

,

b

]

and

[

b

,

c

]

,  i.e. on the whole greater interval

[

a

,

c

]

,  the condition

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

<

ε

.

Title

uniform convergence on union interval

Canonical name

UniformConvergenceOnUnionInterval

Date of creation

2013-03-22 17:27:09

Last modified on

2013-03-22 17:27:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A30

Related topic

MinimalAndMaximalNumber

Generated on Fri Feb 9 21:24:29 2018 by

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Gronwall’s theorem

Gronwall’s theorem

The function

G

(

n

)

:=

σ

⁢

(

n

)

n

⁢

ln

⁡

(

ln

⁡

n

)

(

n

=

2

,

3

,

4

,

…

)

,

in which

σ

⁢

(

n

)

means the sum of the positive divisors of

n

, satisfies the equation

lim sup

n

→

∞

⁡

G

⁢

(

n

)

=

e

γ

where

γ

is the Euler–

Mascheroni constant

.

References

1

T. H. Gronwall

:  Some asymptotic expressions in the theory of numbers.

-

Trans. Amer. Math. Soc.

14

(1913) 113–122.

Title

Gronwall’s theorem

Canonical name

GronwallsTheorem

Date of creation

2013-03-22 19:33:44

Last modified on

2013-03-22 19:33:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A99

Classification

msc 11A25

Classification

msc 26A12

Related topic

EulerMascheroniConstant

Related topic

RobinsTheorem

Generated on Fri Feb 9 21:24:30 2018 by

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proof of Silverman-Toeplitz theorem

proof of Silverman-Toeplitz theorem

First, we shall show that the series

∑

n

=

0

∞

a

m

⁢

n

⁢

z

n

converges

. Since the

sequence

{

z

n

}

converges, it must be

bounded

in

absolute value

— there must exist a constant

K

>

0

such that

|

z

n

|

≤

K

for all

n

. Hence,

|

a

m

⁢

n

⁢

z

n

|

≤

K

⁢

|

a

m

⁢

n

|

.

Summing this gives

∑

n

=

0

∞

|

a

m

⁢

n

⁢

z

n

|

≤

K

⁢

∑

n

=

0

∞

|

a

m

⁢

n

|

≤

K

⁢

B

.

Hence, the series

∑

n

=

0

∞

a

m

⁢

n

⁢

z

n

is

absolutely convergent

which, in turn, implies that it converges.

Let

z

denote the limit of the sequence

{

z

n

}

as

n

→

∞

. Then

|

z

|

≤

K

.
We need to show that, for every

ϵ

>

0

, there exists an integer

M

such that

|

∑

n

=

0

∞

a

m

⁢

n

⁢

z

n

-

z

|

<

ϵ

whenever

m

>

M

.

Since the sequence

{

z

n

}

converges, there must exist an integer

n

1

such that

|

z

n

-

z

|

<

ϵ

4

⁢

B

whenever

n

>

n

1

.

By condition 3, there must exist constants

m

o

,

m

1

,

…

,

m

n

1

such that

|

a

m

⁢

n

|

<

ϵ

4

⁢

(

n

1

+

1

)

⁢

(

K

+

1

)

for

0

≤

n

≤

n

1

and

m

>

m

n

.

(1)

Choose

m

′

=

max

⁡

{

m

0

,

m

1

,

…

,

m

n

1

}

.

Then

|

∑

n

=

0

n

1

a

m

⁢

n

⁢

z

n

|

≤

∑

n

=

0

n

1

|

a

m

⁢

n

⁢

z

n

|

<

∑

n

=

0

n

1

|

z

n

|

⁢

ϵ

4

⁢

(

n

1

+

1

)

⁢

(

K

+

1

)

<

ϵ

4

(2)

when

m

>

m

′

.

By condition 2, there exists a constant

m

′′

such that

|

∑

n

=

0

∞

a

m

⁢

n

-

1

|

<

ϵ

6

⁢

(

|

z

|

+

1

)

whenever

m

>

m

′′

. By (1),

|

∑

n

=

0

n

1

a

m

⁢

n

|

<

ϵ

4

⁢

(

K

+

1

)

≤

ϵ

4

⁢

(

|

z

|

+

1

)

when

m

>

m

′

. Hence, if

m

>

m

′

and

m

>

m

′′

, we have

|

∑

n

=

n

1

+

1

∞

a

m

⁢

n

-

1

|

<

ϵ

2

⁢

(

|

z

|

+

1

)

.

(3)

and

|

∑

n

=

n

1

+

1

∞

a

m

⁢

n

⁢

(

z

n

-

z

)

|

≤

∑

n

=

n

1

+

1

∞

|

a

m

⁢

n

|

⁢

|

z

n

-

z

|

<

ϵ

4

⁢

B

⁢

∑

n

=

0

∞

|

a

m

⁢

n

|

≤

ϵ

4

.

(4)

By the

triangle inequality

and (2), (3), (4) it follows that

|

∑

n

=

0

∞

a

m

⁢

n

⁢

z

n

-

z

|

≤

|

∑

n

=

0

n

1

a

m

⁢

n

⁢

z

n

|

+

|

∑

n

=

n

1

+

1

∞

a

m

⁢

n

⁢

(

z

n

-

z

)

|

+

|

z

|

⁢

|

∑

n

=

n

1

+

1

∞

a

m

⁢

n

-

1

|

<

ϵ

4

+

ϵ

4

+

ϵ

2

=

ϵ

.

Title

proof of Silverman-Toeplitz theorem

Canonical name

ProofOfSilvermanToeplitzTheorem

Date of creation

2013-03-22 14:51:35

Last modified on

2013-03-22 14:51:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

27

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 40B05

Generated on Fri Feb 9 21:24:34 2018 by

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Silverman-Toeplitz theorem

Silverman-Toeplitz theorem

Let

{

a

m

⁢

n

}

be a double sequence of

complex numbers

and let

B

be a positive real number such that:

1.

∑

n

=

0

∞

|

a

m

⁢

n

|

≤

B

for all

m

=

0

,

1

,

2

,

…

2.

lim

m

→

∞

⁡

∑

n

=

0

∞

a

m

⁢

n

=

1

3.

For every

n

=

0

,

1

,

2

,

…

, it is the case that

lim

m

→

∞

⁡

a

m

⁢

n

=

0

Then, if the

sequence

{

z

n

}

converges

, the series

∑

n

=

0

∞

a

m

⁢

n

⁢

z

n

converges and

lim

n

→

∞

⁡

z

n

=

lim

m

→

∞

⁡

∑

n

=

0

∞

a

m

⁢

n

⁢

z

n

Title

Silverman-Toeplitz theorem

Canonical name

SilvermanToeplitzTheorem

Date of creation

2013-03-22 14:51:28

Last modified on

2013-03-22 14:51:28

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 40B05

Generated on Fri Feb 9 21:24:37 2018 by

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arithmetic-geometric series

arithmetic-geometric series

It is well known that a finite

geometric series

is given by

G

n

⁢

(

q

)

=

∑

k

=

1

n

q

k

=

q

1

-

q

⁢

(

1

-

q

n

)

,

q

≠

1

,

(1)

where in general

q

=

r

⁢

e

i

⁢

θ

is

complex

.
When we are dealing with such sums it is common to consider the expression

H

n

⁢

(

q

)

:=

∑

k

=

1

n

k

⁢

q

k

,

q

≠

1

,

(2)

which we shall call an

arithmetic-geometric series

. Let us derive a formula for

H

n

⁢

(

q

)

.

H

n

⁢

(

q

)

=

∑

k

=

1

n

k

⁢

q

k

,

q

⁢

H

n

⁢

(

q

)

=

∑

k

=

1

n

k

⁢

q

k

+

1

.

Subtracting,

(

1

-

q

)

⁢

H

n

⁢

(

q

)

=

∑

k

=

1

n

k

⁢

q

k

-

∑

k

=

1

n

k

⁢

q

k

+

1

=

∑

k

=

1

n

k

⁢

q

k

-

∑

k

=

2

n

+

1

(

k

-

1

)

⁢

q

k

=

∑

k

=

1

n

k

⁢

q

k

-

∑

k

=

2

n

(

k

-

1

)

⁢

q

k

-

n

⁢

q

n

+

1

.

We will proceed to eliminate the right-hand side sums.

(

1

-

q

)

⁢

H

n

⁢

(

q

)

=

q

+

∑

k

=

2

n

q

k

-

n

⁢

q

n

+

1

=

∑

k

=

1

n

q

k

-

n

⁢

q

n

+

1

.

By using (1) and solving for

H

n

⁢

(

q

)

, we obtain

H

n

(

q

)

=

∑

k

=

1

n

k

q

k

=

q

(

1

-

q

)

2

(

1

-

q

n

)

-

n

⁢

q

n

+

1

1

-

q

⋅

(3)

The formula (3) holds in any commutative ring with 1, as long as

(

1

-

q

)

is

invertible

. If

q

is a

complex number

and

|

q

|

<

1

, (3) is the

partial sum

of the

convergent series

H

⁢

(

q

)

=

lim

n

→

∞

⁡

H

n

⁢

(

q

)

=

lim

n

→

∞

⁡

∑

k

=

1

n

k

⁢

q

k

=

lim

n

→

∞

⁡

[

q

(

1

-

q

)

2

⁢

(

1

-

q

n

)

-

n

⁢

q

n

+

1

1

-

q

]

,

that is,

H

⁢

(

q

)

=

∑

k

=

1

∞

k

⁢

q

k

=

q

(

1

-

q

)

2

,

|

q

|

<

1

.

(4)

This last result giving the sum of a converging arithmetic-geometric series may be, naturally, obtained also from the sum formula of the converging geometric series, i.e.

1

+

q

+

q

2

+

q

3

+

…

=

1

1

-

q

,

when one

differentiates

both sides with respect to

q

and then multiplies them by

q

:

1

+

2

⁢

q

+

3

⁢

q

2

+

…

=

1

(

1

-

q

)

2

,

q

+

2

⁢

q

2

+

3

⁢

q

3

+

…

=

q

(

1

-

q

)

2

(A

power series

can be differentiated termwise on the

open interval

of

convergence

.)

Title

arithmetic-geometric series

Canonical name

ArithmeticgeometricSeries

Date of creation

2013-03-22 16:02:15

Last modified on

2013-03-22 16:02:15

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

6

Author

perucho (2192)

Entry type

Derivation

Classification

msc 40C99

Generated on Fri Feb 9 21:24:41 2018 by

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