aarjaneiro/planetmath huggingface dataset:
generalizations of the Leibniz rule

generalizations of the Leibniz rule

For the

derivative

, the

product rule

(

f

⁢

g

)

′

=

f

′

⁢

g

+

f

⁢

g

′

is known as the

Leibniz rule

. Below are various ways it
can be generalized.

Higher derivatives

Let

f

,

g

be real (or complex)

functions

defined on an

open interval

of

ℝ

. If

f

and

g

are

k

times

differentiable

, then

(

f

⁢

g

)

(

k

)

=

∑

r

=

0

k

(

k

r

)

⁢

f

(

k

-

r

)

⁢

g

(

r

)

.

Generalized Leibniz rule for more functions

Let

f

1

,

…

,

f

r

be real (or complex) valued functions
that are defined on an open interval of

ℝ

.
If

f

1

,

…

,

f

r

are

n

times differentiable, then

d

n

d

⁢

t

n

⁢

∏

i

=

1

r

f

i

⁢

(

t

)

=

∑

n

1

+

⋯

+

n

r

=

n

(

n

n

1

,

n

2

,

…

,

n

r

)

⁢

∏

i

=

1

r

d

n

i

d

⁢

t

n

i

⁢

f

i

⁢

(

t

)

.

where

(

n

n

1

,

n

2

,

…

,

n

r

)

is the

multinomial coefficient

.

Leibniz rule for multi-indices

If

f

,

g

:

ℝ

n

→

ℝ

are

smooth functions

defined on an

open set

of

ℝ

n

, and

j

is a

multi-index

, then

∂

j

⁡

(

f

⁢

g

)

=

∑

i

≤

j

(

j

i

)

⁢

∂

i

⁡

(

f

)

⁢

∂

j

-

i

⁡

(

g

)

,

where

i

is a multi-index.

References

1

Leibniz, Gottfried W.

Symbolismus memorabilis calculi Algebraici et Infinitesimalis, in comparatione potentiarum et differentiarum; et de Lege Homogeneorum Transcendentali

, Miscellanea Berolinensia ad incrementum
scientiarum, ex scriptis Societati Regiae scientarum pp. 160-165 (1710).
Available online at the

http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=01-misc/1&seite:int=184

digital library of the
Berlin-Brandenburg Academy.

Title

generalizations of the Leibniz rule

Canonical name

GeneralizationsOfTheLeibnizRule

Date of creation

2013-03-22 14:30:18

Last modified on

2013-03-22 14:30:18

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

13

Author

GeraW (6138)

Entry type

Theorem

Classification

msc 26A06

Synonym

Leibniz rule

Related topic

MultinomialTheorem

Related topic

NthDerivativeOfADeterminant

Generated on Fri Feb 9 19:43:37 2018 by

LaTeXML
generalized Darboux function

generalized Darboux function

Recall that a function

f

:

I

→

ℝ

, where

I

⊆

ℝ

is an

interval

, is called a

Darboux function

if it satisfies the

intermediate value theorem

. This means, that if

a

,

b

∈

I

and

f

⁢

(

a

)

⩽

d

⩽

f

⁢

(

b

)

for some

d

∈

ℝ

, then there exists

c

∈

I

such that

a

⩽

c

⩽

b

and

f

⁢

(

c

)

=

d

.

Darboux proved (see parent object) that if

f

:

[

a

,

b

]

→

ℝ

is

differentiable

then

f

′

is a Darboux function. The class of Darboux functions is very wide. It can be shown that any function

f

:

ℝ

→

ℝ

can be written as a sum of two Darboux functions. We wish to give more general definiton of Darboux function.

Definition.

Let

X

,

Y

be

topological spaces

. Function

f

:

X

→

Y

is called a

(generalized) Darboux function

if and only if whenever

C

⊆

X

is a

connected

subset, then so is

f

⁢

(

C

)

⊆

Y

.

It can be easily proved that connected subsets of intervals (in

ℝ

) are exactly intervals. Thus this definition coincides with classical definiton, when

X

is an interval and

Y

=

ℝ

.

Note that every

continuous map

is a Darboux function.

Also the composition of Darboux functions is again a Darboux function and thus the class of all topological spaces, together with Darboux functions forms a category. The category of topological spaces and continuous maps is its subcategory.

Title

generalized Darboux function

Canonical name

GeneralizedDarbouxFunction

Date of creation

2013-03-22 19:18:36

Last modified on

2013-03-22 19:18:36

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 26A06

Generated on Fri Feb 9 19:43:38 2018 by

LaTeXML
generalized intermediate value theorem

generalized intermediate value theorem

Theorem

.

Let

f

:

X

→

Y

be a

continuous function

with

X

a

connected space

and

Y

a

totally ordered set

in the

order topology

. If

x

1

,

x

2

∈

X

and

y

∈

Y

lies between

f

⁢

(

x

1

)

and

f

⁢

(

x

2

)

, then there exists

x

∈

X

such that

f

⁢

(

x

)

=

y

.

Proof.

The sets

U

=

f

⁢

(

X

)

∩

(

-

∞

,

y

)

and

V

=

f

⁢

(

X

)

∩

(

y

,

∞

)

are

disjoint

open subsets

of

f

⁢

(

X

)

in the

subspace topology

, and they are both non-empty, as

f

⁢

(

x

1

)

is

contained

in one and

f

⁢

(

x

2

)

is contained in the other. If

y

∉

f

⁢

(

X

)

, then

U

∪

V

constitutes a of the space

f

⁢

(

X

)

, contradicting the

hypothesis

that

f

⁢

(

X

)

is the

continuous

image of the connected space

X

. Thus there must exist

x

∈

X

such that

f

⁢

(

x

)

=

y

.
∎

This version of the

intermediate value theorem

reduces to the familiar one of

http://planetmath.org/node/7599

real analysis

when

X

is taken to be a

closed interval

in

ℝ

and

Y

is taken to be

ℝ

.

References

1

J. Munkres,

Topology

, 2nd ed. Prentice Hall, 1975.

Title

generalized intermediate value theorem

Canonical name

GeneralizedIntermediateValueTheorem

Date of creation

2013-03-22 17:17:44

Last modified on

2013-03-22 17:17:44

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

8

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 26A06

Related topic

OrderTopology

Related topic

TotalOrder

Related topic

Continuous

Related topic

ConnectedSpace

Related topic

ConnectednessIsPreservedUnderAContinuousMap

Generated on Fri Feb 9 19:43:40 2018 by

LaTeXML
harmonic number

harmonic number

The

harmonic number of order

n

of

θ

is defined as

H

θ

⁢

(

n

)

=

∑

i

=

1

n

1

i

θ

Note that

n

may be equal to

∞

, provided

θ

>

1

.

If

θ

≤

1

, while

n

=

∞

, the

harmonic series

does not

converge

and hence the

harmonic number

does not exist.

If

θ

=

1

, we may just write

H

θ

⁢

(

n

)

as

H

n

(this is a common notation).

•

If

ℜ

⁡

(

θ

)

>

1

and

n

=

∞

then the sum is the

Riemann zeta function

.

•

If

θ

=

1

, then we get what is known simply as“the harmonic number”, and it has many important properties. For example, it has

asymptotic expansion

H

n

=

ln

⁡

n

+

γ

+

1

2

⁢

m

+

…

where

γ

is Euler’s constant.

•

It is possible

1

1

See “The Art of computer programming” vol. 2 by D. Knuth

to define harmonic numbers for non-integral

n

. This is done by means of the series

H

n

⁢

(

z

)

=

∑

n

≥

1

(

n

-

z

-

(

n

+

x

)

-

z

)

.

Title

harmonic number

Canonical name

HarmonicNumber

Date of creation

2013-03-22 13:01:28

Last modified on

2013-03-22 13:01:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 26A06

Classification

msc 40A05

Related topic

Series

Related topic

AbsoluteConvergence

Related topic

HarmonicSeries

Related topic

PrimeHarmonicSeries

Related topic

WolstenholmesTheorem

Defines

harmonic number of order

Generated on Fri Feb 9 19:43:42 2018 by

LaTeXML
improper limits

improper limits

In

calculus

there is often used such expressions as “the limit of a

function

is

infinite

”, and one may write for instance that

lim

x

→

0

⁡

1

x

2

=

∞

.

Such “limits” are actually of the limit notion, and can be defined exactly.  They are called

improper limits

.

Definition.

Let the

real function

f

be defined in a neighbourhood of the point

x

0

.

lim

x

→

x

0

⁡

f

⁢

(

x

)

=

∞

iff for every

real number

M

there exists a number

δ

M

such that

f

⁢

(

x

)

>

M

as soon as

0

<

|

x

-

x

0

|

<

δ

M

.

In a similar way we can define the improper limit

-

∞

of a real function.  The definition may be extended also to the cases

x

→

±

∞

, when one speaks of

limits at infinity

.

Note 1.

If

lim

x

→

x

0

⁡

f

⁢

(

x

)

=

∞

and

lim

x

→

x

0

⁡

g

⁢

(

x

)

=

a

>

0

,  then we have

lim

x

→

x

0

⁡

f

⁢

(

x

)

⁢

g

⁢

(

x

)

=

∞

.

Hence we can say that

∞

⋅

a

=

∞

when

a

>

0

.  There are some other “mnemonics of infinite” (cf. the extended real numbers):

∞

⋅

a

=

-

∞

(

a

<

0

)

±

∞

+

a

=

±

∞

a

±

∞

=

0

∞

+

∞

=

∞

∞

⋅

∞

=

∞

-

∞

⋅

∞

=

-

∞

On the contrary, there exist no mnemonics for the cases

∞

⋅

0

,

∞

-

∞

,

∞

∞

,

0

0

,

0

0

,

∞

0

,

1

∞

;

they are and depend on the instance (cf. the

indeterminate form

).

Note 2.

In the

complex plane

, the expression

lim

z

→

z

0

⁡

f

⁢

(

z

)

=

∞

means that

lim

z

→

z

0

⁡

|

f

⁢

(

z

)

|

=

∞

.

Title

improper limits

Canonical name

ImproperLimits

Date of creation

2013-03-22 14:40:45

Last modified on

2013-03-22 14:40:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A06

Synonym

infinite limits

Synonym

improper limit

Related topic

LHpitalsRule

Related topic

ExtendedRealNumbers

Related topic

LimitRulesOfFunctions

Related topic

IntegratingTanXOver0fracpi2

Related topic

IndeterminateForm

Related topic

ExampleOfJumpDiscontinuity

Related topic

ListOfCommonLimits

Related topic

LimitsOfNaturalLogarithm

Related topic

SecondDerivativeAsSimpleLimit

Related topic

AngleBetweenTwoLines

Defines

limit at infinity

Defines

mnemonic of infinite

Generated on Fri Feb 9 19:43:45 2018 by

LaTeXML
increasing/decreasing/monotone function

increasing/decreasing/monotone function

Definition

Let

A

be a subset of

ℝ

, and let

f

be a

function

from

f

:

A

→

ℝ

.
Then

1.

f

is

increasing

or

weakly increasing

, if

x

≤

y

implies that

f

⁢

(

x

)

≤

f

⁢

(

y

)

(for all

x

and

y

in

A

).

2.

f

is

strictly increasing

or

strongly increasing

, if

x

<

y

implies that

f

⁢

(

x

)

<

f

⁢

(

y

)

.

3.

f

is

decreasing

or

weakly decreasing

, if

x

≤

y

implies that

f

⁢

(

x

)

≥

f

⁢

(

y

)

.

4.

f

is

strictly decreasing

or

strongly decreasing

if

x

<

y

implies that

f

⁢

(

x

)

>

f

⁢

(

y

)

.

5.

f

is

monotone

,
if

f

is either increasing or decreasing.

6.

f

is

strictly monotone

or

strongly monotone

,
if

f

is either strictly increasing or strictly decreasing.

Theorem

Let

X

be a

bounded

or

unbounded

open interval

of

ℝ

.
In other words, let

X

be an interval of the form

X

=

(

a

,

b

)

, where

a

,

b

∈

ℝ

∪

{

-

∞

,

∞

}

.
Futher, let

f

:

X

→

ℝ

be a monotone function.

1.

The set of points where

f

is

discontinuous

is at most

countable

[

1

,

2

]

.

Lebesgue

f

is

differentiable

almost
everywhere (

[

3

]

, pp. 514).

References

1

C.D. Aliprantis, O. Burkinshaw,

Principles of

Real Analysis

,
2nd ed., Academic Press, 1990.

2

W. Rudin,

Principles of

Mathematical Analysis

, McGraw-Hill Inc., 1976.

3

F. Jones,

Lebesgue Integration on

Euclidean Spaces

,
Jones and Barlett Publishers, 1993.

Title

increasing/decreasing/monotone function

Canonical name

IncreasingdecreasingmonotoneFunction

Date of creation

2013-03-22 13:36:05

Last modified on

2013-03-22 13:36:05

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

12

Author

Koro (127)

Entry type

Definition

Classification

msc 26A06

Classification

msc 26A48

Defines

increasing

Defines

decreasing

Defines

strictly increasing

Defines

strictly decreasing

Defines

monotone

Defines

monotonic

Defines

strictly monotone

Defines

strictly monotonic

Defines

weakly increasing

Defines

weakly decreasing

Defines

strongly increasing

Defines

strongly decreasing

Defines

strongly monotone

Defines

weakly monotone

Defines

stronly mono

Generated on Fri Feb 9 19:43:47 2018 by

LaTeXML
integral mean value theorem

integral mean value theorem

The Integral Mean Value Theorem

.

If

f

and

g

are

continuous

real functions on an

interval

[

a

,

b

]

, and

g

is additionally non-negative on

(

a

,

b

)

, then there exists a

ζ

∈

(

a

,

b

)

such that

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

d

x

=

f

⁢

(

ζ

)

⁢

∫

a

b

g

⁢

(

x

)

⁢

d

x

.

Proof.

Since

f

is continuous on a closed

bounded set

,

f

is

bounded

and attains its bounds, say

f

⁢

(

x

0

)

≤

f

⁢

(

x

)

≤

f

⁢

(

x

1

)

for all

x

∈

[

a

,

b

]

. Thus, since

g

is non-negative for all

x

∈

[

a

,

b

]

f

⁢

(

x

0

)

⁢

g

⁢

(

x

)

≤

f

⁢

(

x

)

⁢

g

⁢

(

x

)

≤

f

⁢

(

x

1

)

⁢

g

⁢

(

x

)

.

Integrating both sides gives

f

⁢

(

x

0

)

⁢

∫

a

b

g

⁢

(

x

)

⁢

d

x

≤

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

d

x

≤

f

⁢

(

x

1

)

⁢

∫

a

b

g

⁢

(

x

)

⁢

d

x

.

If

∫

a

b

g

⁢

(

x

)

⁢

d

x

=

0

, then

g

⁢

(

x

)

is identically zero, and the result follows trivially. Otherwise,

f

⁢

(

x

0

)

≤

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

d

x

∫

a

b

g

⁢

(

x

)

⁢

d

x

≤

f

⁢

(

x

1

)

,

and the result follows from the

intermediate value theorem

.
∎

Title

integral mean value theorem

Canonical name

IntegralMeanValueTheorem

Date of creation

2013-03-22 17:15:56

Last modified on

2013-03-22 17:15:56

Owner

me_and (17092)

Last modified by

me_and (17092)

Numerical id

9

Author

me_and (17092)

Entry type

Theorem

Classification

msc 26A06

Related topic

EstimatingTheoremOfContourIntegral

Generated on Fri Feb 9 19:43:49 2018 by

LaTeXML
integrals of even and odd functions

integrals of even and odd functions

Theorem

.

Let the

real function

f

be
Riemann-integrable (

http://planetmath.org/RiemannIntegrable

) on

[

-

a

,

a

]

.  If

f

is an

•

even function

, then

∫

-

a

a

f

⁢

(

x

)

⁢

𝑑

x

=

2

⁢

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

,

•

odd function

, then

∫

-

a

a

f

⁢

(

x

)

⁢

𝑑

x

=

0

.

Of course, both cases concern the

zero map

which is both
.

Proof.

Since the

definite integral

is additive with respect to the interval of integration, one has

I

:=

∫

-

a

a

f

⁢

(

x

)

⁢

𝑑

x

=

∫

-

a

0

f

⁢

(

t

)

⁢

𝑑

t

+

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

.

Making in the first addend the substitution

t

=

-

x

,

d

⁢

t

=

-

d

⁢

x

and swapping the limits of integration one gets

I

=

∫

a

0

f

⁢

(

-

x

)

⁢

(

-

d

⁢

x

)

+

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

=

∫

0

a

f

⁢

(

-

x

)

⁢

𝑑

x

+

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

.

Using then the

definitions

of even (

http://planetmath.org/EvenoddFunction

) (

+

) and odd (

http://planetmath.org/EvenoddFunction

) (

-

)

function

yields

I

=

∫

0

a

(

±

f

⁢

(

x

)

)

⁢

𝑑

x

+

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

=

±

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

+

∫

0

a

f

⁢

(

x

)

⁢

𝑑

x

,

which settles the equations of the theorem.

Title

integrals of even and odd functions

Canonical

name

IntegralsOfEvenAndOddFunctions

Date of creation

2014-03-13 16:17:44

Last modified on

2014-03-13 16:17:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Synonym

integral of odd function

Synonym

integral of even function

Related topic

DefiniteIntegral

Related topic

ChangeOfVariableInDefiniteIntegral

Related topic

ExampleOfUsingResidueTheorem

Related topic

FourierSineAndCosineSeries

Related topic

IntegralOverAPeriodInterval

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intermediate value theorem

intermediate value theorem

If

f

is a real-valued

continuous function

on the

interval

[

a

,

b

]

,
and

x

1

and

x

2

are points with

a

≤

x

1

<

x

2

≤

b

such that

f

⁢

(

x

1

)

≠

f

⁢

(

x

2

)

,
then for every

y

strictly between

f

⁢

(

x

1

)

and

f

⁢

(

x

2

)

there is a

c

∈

(

x

1

,

x

2

)

such that

f

⁢

(

c

)

=

y

.

Bolzano’s

theorem

is a special case of this.

The theorem can be generalized as follows:
If

f

is a real-valued continuous function
on a

connected

topological space

X

,
and

x

1

,

x

2

∈

X

with

f

⁢

(

x

1

)

≠

f

⁢

(

x

2

)

,
then for every

y

between

f

⁢

(

x

1

)

and

f

⁢

(

x

2

)

there is a

ξ

∈

X

such that

f

⁢

(

ξ

)

=

y

.
(However, this “

generalization

” is essentially

trivial

,
and in order to derive the

intermediate value theorem

from it
one must first establish the less trivial fact that

[

a

,

b

]

is connnected.)
This result remains true
if the

codomain

is an arbitrary

ordered set

with its

order topology

;
see the entry
proof of

generalized intermediate value theorem

(

http://planetmath.org/ProofOfGeneralizedIntermediateValueTheorem

)
for a proof.

Title

intermediate value theorem

Canonical name

IntermediateValueTheorem

Date of creation

2013-03-22 11:51:29

Last modified on

2013-03-22 11:51:29

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 70F25

Classification

msc 17B50

Classification

msc 81-00

Related topic

RollesTheorem

Related topic

MeanValueTheorem

Related topic

Continuous

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intermediate value theorem for extended real numbers

intermediate value theorem for extended real numbers

Theorem 1

.

Let

R

¯

be the

extended real numbers

, and
suppose

f

:

R

¯

→

R

¯

is a

continuous function

.
Suppose

x

1

<

x

2

∈

R

¯

are such that

f

⁢

(

x

1

)

≠

f

⁢

(

x

2

)

. If

y

∈

(

f

⁢

(

x

1

)

,

f

⁢

(

x

2

)

)

, then
for some

c

∈

(

x

1

,

x

2

)

we have

f

⁢

(

c

)

=

y

.

Proof.

As

ℝ

¯

is

homeomorphic

to

[

0

,

1

]

, we can assume that

f

is a

function

f

:

[

0

,

1

]

→

ℝ

¯

. For simplicity,
let us also assume that

x

1

=

0

,

x

2

=

1

, and

f

⁢

(

0

)

<

f

⁢

(

1

)

. Then
for some

ε

>

0

we have

f

⁢

(

0

)

<

y

-

ε

<

y

<

y

+

ε

<

f

⁢

(

1

)

.

Let

g

:

[

0

,

1

]

→

ℝ

be the continuous function

g

⁢

(

x

)

=

max

⁡

{

min

⁡

{

f

⁢

(

x

)

,

y

+

ε

}

,

y

-

ε

}

.

Now

g

⁢

(

0

)

=

y

-

ε

and

g

⁢

(

1

)

=

y

+

ε

,
so for some

c

∈

(

0

,

1

)

, we have

g

⁢

(

c

)

=

y

, and thus

f

⁢

(

c

)

=

y

.
∎

Title

intermediate value theorem for extended real numbers

Canonical name

IntermediateValueTheoremForExtendedRealNumbers

Date of creation

2013-03-22 15:35:15

Last modified on

2013-03-22 15:35:15

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 26A06

Related topic

ExtendedRealNumbers

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Kummer’s acceleration method

Kummer’s acceleration method

There are several methods for acceleration of the convergence of a given series

∑

n

=

1

∞

a

n

=

S

.

(1)

One of the simplest is the following one due to Kummer (1837).

We suppose that the terms

a

n

of (1) are nonzero.  Let

∑

n

=

1

∞

b

n

=

C

be a series with nonzero terms and the known sum

C

.  We use the limit

lim

n

→

∞

⁡

a

n

b

n

=

ϱ

≠

0

and the identity

S

=

ϱ

⁢

C

+

∑

n

=

1

∞

(

1

-

ϱ

⁢

b

n

a

n

)

⁢

a

n

.

(2)

Thus the original series (1) has attained a new form (2) the convergence of which is faster because of

lim

n

→

∞

⁡

(

1

-

ϱ

⁢

b

n

a

n

)

=

0

.

Example.

For replacing the series

∑

n

=

1

∞

1

n

2

=

S

by a faster converging series we may take

∑

n

=

1

∞

1

n

⁢

(

n

+

1

)

=

:

C

,

which, for its part, can be expressed as the telescoping series

C

=

∑

n

=

1

∞

(

1

n

-

1

n

+

1

)

=

1

.

Now we have

ϱ

=

1

,  and using (2) we obtain

S

=

1

+

∑

n

=

1

∞

1

n

2

⁢

(

n

+

1

)

.

The convergence of this series may accelerated similarly taking e.g.

∑

n

=

1

∞

1

n

⁢

(

n

+

1

)

⁢

(

n

+

2

)

=

:

C

,

where now

C

=

1

4

;  then we get

S

=

5

4

+

2

⁢

∑

n

=

1

∞

1

n

2

⁢

(

n

+

1

)

⁢

(

n

+

2

)

.

The procedure may be repeated

N

times in all, yielding the result

S

=

∑

n

=

1

N

1

n

2

+

N

!

⁢

∑

n

=

1

∞

1

n

2

⁢

(

n

+

1

)

⁢

(

n

+

2

)

⁢

⋯

⁢

(

n

+

N

)

.

As for the sum of this series, see

Riemann zeta function

at

s

=

2

(

http://planetmath.org/valueoftheriemannzetafunctionats2

).

References

1

Pascal Sebah & Xavier Gourdon

:

http://numbers.

computation

.free.fr/Constants/constants.html

Acceleration of the convergence of series

(2002).

Title

Kummer’s acceleration method

Canonical name

KummersAccelerationMethod

Date of creation

2014-12-12 10:34:19

Last modified on

2014-12-12 10:34:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 26A06

Classification

msc 40A05

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Generated on Fri Feb 9 19:43:59 2018 by

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limit

limit

Let

X

and

Y

be

metric spaces

and let

a

∈

X

be a

limit point

of

X

. Suppose that

f

:

X

∖

{

a

}

→

Y

is a

function

defined everywhere except at

a

. For

L

∈

Y

, we say the

limit

of

f

⁢

(

x

)

as

x

approaches

a

is equal to

L

, or

lim

x

→

a

⁡

f

⁢

(

x

)

=

L

if, for every

real number

ε

>

0

, there exists a real
number

δ

>

0

such that, whenever

x

∈

X

with

0

<

d

X

⁢

(

x

,

a

)

<

δ

, then

d

Y

⁢

(

f

⁢

(

x

)

,

L

)

<

ε

.

The formal definition of limit as given above has a well–deserved
reputation for being notoriously hard for inexperienced students to
master. There is no easy fix for this problem, since the

concept

of a
limit is inherently difficult to state precisely (and indeed wasn’t
even accomplished historically until the 1800’s by Cauchy, well after
the

development

of calculus in the 1600’s by Newton and Leibniz).
However, there are number of related definitions, which, taken
together, may shed some light on the nature of the concept.

•

The notion of a limit can be generalized to

mappings

between arbitrary

topological spaces

, under some mild

restrictions

. In this context we say that

lim

x

→

a

⁡

f

⁢

(

x

)

=

L

if

a

is a limit point of

X

and, for every

neighborhood

V

of

L

(in

Y

), there is a deleted neighborhood

U

of

a

(in

X

) which is mapped into

V

by

f

. One also requires that the range

Y

be

Hausdorff

(or at least

T

1

) in order to ensure that limits, when they exist, are unique.

•

Let

a

n

,

n

∈

ℕ

be a

sequence

of

elements

in a metric
space

X

. We say that

L

∈

X

is the limit of the sequence, if
for every

ε

>

0

there exists a

natural number

N

such
that

d

⁢

(

a

n

,

L

)

<

ε

for all natural numbers

n

>

N

.

•

The definition of the limit of a mapping can be based on the
limit of a sequence. To wit,

lim

x

→

a

⁡

f

⁢

(

x

)

=

L

if and only
if, for every sequence of points

x

n

in

X

converging to

a

(that is,

x

n

→

a

,

x

n

≠

a

), the sequence of points

f

⁢

(

x

n

)

in

Y

converges

to

L

.

In calculus,

X

and

Y

are frequently taken to be

Euclidean spaces

ℝ

n

and

ℝ

m

, in which case the distance
functions

d

X

and

d

Y

cited above are just

Euclidean distance

.

Title

limit

Canonical name

Limit

Date of creation

2013-03-22 12:28:25

Last modified on

2013-03-22 12:28:25

Owner

djao (24)

Last modified by

djao (24)

Numerical id

12

Author

djao (24)

Entry type

Definition

Classification

msc 26A06

Classification

msc 26B12

Classification

msc 54E35

Related topic

Continuous

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