aarjaneiro/planetmath huggingface dataset:
fundamental theorems of calculus for Lebesgue integration

fundamental theorems of calculus for Lebesgue integration

Loosely, the

Fundamental Theorems of Calculus

serve to demonstrate that integration and

differentiation

are

inverse

processes. Suppose that

F

⁢

(

x

)

is an

absolutely continuous function

on an

interval

[

a

,

b

]

⊂

ℝ

. The two following forms of the

theorem

are

equivalent

.

First form of the Fundamental Theorem:

There exists a

function

f

⁢

(

t

)

Lebesgue-integrable on

[

a

,

b

]

such that for any

x

∈

[

a

,

b

]

, we have

F

⁢

(

x

)

-

F

⁢

(

a

)

=

∫

a

x

f

⁢

(

t

)

⁢

𝑑

t

.

Second form of the Fundamental Theorem:

F

⁢

(

x

)

is

differentiable

almost everywhere on

[

a

,

b

]

and its

derivative

, denoted

F

′

⁢

(

x

)

, is Lebesgue-integrable on that interval. In

addition

, we have the

relation

F

⁢

(

x

)

-

F

⁢

(

a

)

=

∫

a

x

F

′

⁢

(

t

)

⁢

𝑑

t

for any

x

∈

[

a

,

b

]

.

Title

fundamental theorems of calculus for Lebesgue integration

Canonical name

FundamentalTheoremsOfCalculusForLebesgueIntegration

Date of creation

2013-03-22 12:27:54

Last modified on

2013-03-22 12:27:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

17

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 26-00

Synonym

first fundamental theorem of calculus

Synonym

second fundamental theorem of calculus

Synonym

fundamental theorem of calculus

Related topic

FundamentalTheoremOfCalculusClassicalVersion

Related topic

FundamentalTheoremOfCalculusForRiemannIntegration

Related topic

ChangeOfVariableInDefiniteIntegral

Generated on Fri Feb 9 19:40:53 2018 by

LaTeXML
generalized mean

generalized mean

Definition

Let

x

1

,

x

2

,

…

,

x

n

be

real numbers

, and

f

a

continuous

and

strictly increasing

or

decreasing function

on the real
numbers. If each number

x

i

is assigned a weight

p

i

, with

∑

i

=

1

n

p

i

=

1

, for

i

=

1

,

…

,

n

, then the

generalized mean

is defined as

f

-

1

⁢

(

∑

i

=

1

n

p

i

⁢

f

⁢

(

x

i

)

)

.

Special cases

1.

f

⁢

(

x

)

=

x

,

p

i

=

1

/

n

for all

i

:

arithmetic mean

2.

f

⁢

(

x

)

=

x

:

weighted mean

3.

f

⁢

(

x

)

=

log

⁡

(

x

)

,

p

i

=

1

/

n

for all

i

:

geometric mean

4.

f

⁢

(

x

)

=

1

/

x

and

p

i

=

1

/

n

for all

i

:

harmonic mean

5.

f

⁢

(

x

)

=

x

2

and

p

i

=

1

/

n

for all

i

:

root-mean-square

6.

f

⁢

(

x

)

=

x

d

and

p

i

=

1

/

n

for all

i

:

power mean

7.

f

⁢

(

x

)

=

x

d

:

weighted power mean

8.

f

⁢

(

x

)

=

2

(

1

-

α

)

⁢

x

,

α

≠

1

,

x

i

=

-

log

2

⁡

p

i

:
Rényi’s

α

-

entropy

Title

generalized mean

Canonical

name

GeneralizedMean

Date of creation

2013-03-22 14:32:12

Last modified on

2013-03-22 14:32:12

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26-00

Synonym

Kolmogorov-Nagumo

function

of the mean

Synonym

Hölder mean

Generated on Fri Feb 9 19:40:55 2018 by

LaTeXML
mollifier

mollifier

A

mollifier

is a

smooth function

f

:

ℝ

n

→

ℝ

such

1.

f

⁢

(

x

)

≥

0

for all

x

∈

ℝ

n

,

2.

f

is normalized such that

∫

ℝ

n

f

⁢

(

x

)

⁢

𝑑

x

=

1

,

3.

f

⁢

(

0

)

=

1

.

Title

mollifier

Canonical name

Mollifier

Date of creation

2013-03-22 15:01:58

Last modified on

2013-03-22 15:01:58

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 26-00

Generated on Fri Feb 9 19:40:59 2018 by

LaTeXML
proof of the fundamental theorem of calculus

proof of the fundamental theorem of calculus

Recall that a

continuous function

is

Riemann integrable

on every

interval

[

c

,

x

]

, so the

integral

F

⁢

(

x

)

=

∫

c

x

f

⁢

(

t

)

⁢

𝑑

t

is well defined.

Consider the increment of

F

:

F

⁢

(

x

+

h

)

-

F

⁢

(

x

)

=

∫

c

x

+

h

f

⁢

(

t

)

⁢

𝑑

t

-

∫

c

x

f

⁢

(

t

)

⁢

𝑑

t

=

∫

x

x

+

h

f

⁢

(

t

)

⁢

𝑑

t

(we have used the linearity of the integral with respect to the

function

and the

additivity

with respect to the domain).

Since

f

is

continuous

, by the

mean-value theorem

, there exists

ξ

h

∈

[

x

,

x

+

h

]

such that

f

⁢

(

ξ

h

)

=

F

⁢

(

x

+

h

)

-

F

⁢

(

x

)

h

so that

F

′

⁢

(

x

)

=

lim

h

→

0

⁡

F

⁢

(

x

+

h

)

-

F

⁢

(

x

)

h

=

lim

h

→

0

⁡

f

⁢

(

ξ

h

)

=

f

⁢

(

x

)

since

ξ

h

→

x

as

h

→

0

.
This proves the first part of the

theorem

.

For the second part suppose that

G

is any

antiderivative

of

f

, i.e.

G

′

=

f

.
Let

F

be the integral function

F

⁢

(

x

)

=

∫

a

x

f

⁢

(

t

)

⁢

𝑑

t

.

We have just proven that

F

′

=

f

. So

F

′

⁢

(

x

)

=

G

′

⁢

(

x

)

for all

x

∈

[

a

,

b

]

or, which is the same,

(

G

-

F

)

′

=

0

. This means that

G

-

F

is

constant

on

[

a

,

b

]

that is, there exists

k

such that

G

⁢

(

x

)

=

F

⁢

(

x

)

+

k

. Since

F

⁢

(

a

)

=

0

we have

G

⁢

(

a

)

=

k

and hence

G

⁢

(

x

)

=

F

⁢

(

x

)

+

G

⁢

(

a

)

for all

x

∈

[

a

,

b

]

.
Thus

∫

a

b

f

⁢

(

t

)

⁢

𝑑

t

=

F

⁢

(

b

)

=

G

⁢

(

b

)

-

G

⁢

(

a

)

.

Title

proof of the

fundamental theorem of calculus

Canonical name

ProofOfTheFundamentalTheoremOfCalculus

Date of creation

2013-03-22 13:45:37

Last modified on

2013-03-22 13:45:37

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

10

Author

paolini (1187)

Entry type

Proof

Classification

msc 26-00

Generated on Fri Feb 9 19:41:01 2018 by

LaTeXML
root-mean-square

root-mean-square

If

x

1

,

x

2

,

…

,

x

n

are

real numbers

, we define their

root-mean-square

or

quadratic mean

as

R

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

x

1

2

+

x

2

2

+

⋯

+

x

n

2

n

.

The root-mean-square of a

random variable

X is defined as the

square
root

of the

expectation

of

X

2

:

R

⁢

(

X

)

=

E

⁢

(

X

2

)

If

X

1

,

X

2

,

…

,

X

n

are random variables with

standard deviations

σ

1

,

σ

2

,

…

,

σ

n

, then the standard deviation of
their

arithmetic mean

,

X

1

+

X

2

+

⋯

+

X

n

n

, is the
root-mean-square of

σ

1

,

σ

2

,

…

,

σ

n

.

Title

root-mean-square

Canonical

name

Rootmeansquare

Date of creation

2013-03-22 13:10:24

Last modified on

2013-03-22 13:10:24

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

4

Author

pbruin (1001)

Entry type

Definition

Classification

msc 26-00

Classification

msc 26D15

Synonym

root mean square

Synonym

rms

Synonym

quadratic mean

Related topic

ArithmeticMean

Related topic

GeometricMean

Related topic

HarmonicMean

Related topic

PowerMean

Related topic

WeightedPowerMean

Related topic

ArithmeticGeometricMeansInequality

Related topic

GeneralMeansInequality

Related topic

ProofOfGeneralMeansInequality

Generated on Fri Feb 9 19:41:03 2018 by

LaTeXML
stair function

stair function

A

stair

function

is a real function of the form

ϕ

=

∑

k

=

1

n

a

k

⁢

χ

I

k

(1)

where

a

k

∈

ℝ

,

χ

I

k

is the

characteristic function

,

k

=

1

,

2

,

…

,

n

and

I

1

,

I

2

,

…

⁢

I

n

are bounded,

pairwise disjoint

intervals

,
for some

n

∈

ℕ

.

Title

stair function

Canonical name

StairFunction

Date of creation

2013-03-22 15:32:32

Last modified on

2013-03-22 15:32:32

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

6

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 26-00

Generated on Fri Feb 9 19:41:05 2018 by

LaTeXML
table of natural logarithms

table of natural logarithms

This table is for integers in the range

-

1

<

n

<

100

. To find the

natural logarithm

(that is, with base

e

, the

natural log base

) for the desired

n

, look for the ten’s place digit (0 in the case of

-

1

<

n

<

10

) in the leftmost column and the one’s place digit in the topmost row. The

intersection

x

of that row and column thus gives

log

⁡

n

to four or five

decimal places

.

n

0

1

2

3

4

5

6

7

8

9

0

-

∞

0.0000

0.69315

1.09861

1.38629

1.60944

1.79176

1.94591

2.07944

2.19722

1

2.30259

2.3979

2.48491

2.56495

2.63906

2.70805

2.77259

2.83321

2.89037

2.94444

2

2.99573

3.04452

3.09104

3.13549

3.17805

3.21888

3.2581

3.29584

3.3322

3.3673

3

3.4012

3.43399

3.46574

3.49651

3.52636

3.55535

3.58352

3.61092

3.63759

3.66356

4

3.68888

3.71357

3.73767

3.7612

3.78419

3.80666

3.82864

3.85015

3.8712

3.89182

5

3.91202

3.93183

3.95124

3.97029

3.98898

4.00733

4.02535

4.04305

4.06044

4.07754

6

4.09434

4.11087

4.12713

4.14313

4.15888

4.17439

4.18965

4.20469

4.21951

4.23411

7

4.2485

4.26268

4.27667

4.29046

4.30407

4.31749

4.33073

4.34381

4.35671

4.36945

8

4.38203

4.39445

4.40672

4.41884

4.43082

4.44265

4.45435

4.46591

4.47734

4.48864

9

4.49981

4.51086

4.52179

4.5326

4.54329

4.55388

4.56435

4.57471

4.58497

4.59512

Title

table of natural logarithms

Canonical

name

TableOfNaturalLogarithms

Date of creation

2013-03-22 18:08:39

Last modified on

2013-03-22 18:08:39

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 26-00

Classification

msc 26A09

Classification

msc 26A06

Generated on Fri Feb 9 19:41:07 2018 by

LaTeXML
binomial proof of positive integer power rule

binomial proof of positive integer power rule

We will use the difference quotient in this proof of the

power rule

for

positive integers

. Let

f

⁢

(

x

)

=

x

n

for some

integer

n

≥

0

. Then we have

f

′

⁢

(

x

)

=

lim

h

→

0

⁡

(

x

+

h

)

n

-

x

n

h

.

We can use the

binomial theorem

to expand the

numerator

f

′

⁢

(

x

)

=

lim

h

→

0

⁡

C

0

n

⁢

x

0

⁢

h

n

+

C

1

n

⁢

x

1

⁢

h

n

-

1

+

⋯

+

C

n

-

1

n

⁢

x

n

-

1

⁢

h

1

+

C

n

n

⁢

x

n

⁢

h

0

-

x

n

h

where

C

k

n

=

n

!

k

!

⁢

(

n

-

k

)

!

. We can now simplify the above

f

′

⁢

(

x

)

=

lim

h

→

0

⁡

h

n

+

n

⁢

x

⁢

h

n

-

1

+

⋯

+

n

⁢

x

n

-

1

⁢

h

+

x

n

-

x

n

h

=

lim

h

→

0

⁡

(

h

n

-

1

+

n

⁢

x

⁢

h

n

-

2

+

⋯

+

n

⁢

x

n

-

1

)

=

n

⁢

x

n

-

1

=

n

⁢

x

n

-

1

.

Title

binomial proof of positive integer power rule

Canonical name

BinomialProofOfPositiveIntegerPowerRule

Date of creation

2013-03-22 12:29:43

Last modified on

2013-03-22 12:29:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Proof

Classification

msc 26A03

Generated on Fri Feb 9 19:41:10 2018 by

LaTeXML
comparison of s⁢i⁢nθ and θ near θ=0

comparison of

s

⁢

i

⁢

n

θ

and

θ

near

θ

=

0

Theorem 1

.

Let

0

<

θ

<

π

2

, where

θ

is an angle measured in radians. Then

sin

⁡

θ

<

θ

.

Proof.

Let

O

=

(

0

,

0

)

,

P

=

(

1

,

0

)

, and

Q

=

(

cos

⁡

θ

,

sin

⁡

θ

)

. Note that the circle

x

2

+

y

2

=

1

passes through

P

and

Q

and that the shortest arc along this circle from

P

to

Q

has length

θ

. Note also that the

line segments

O

⁢

P

¯

and

O

⁢

Q

¯

are radii of the circle

x

2

+

y

2

=

1

and therefore must each have length 1.

Draw the line segment

P

⁢

Q

¯

. Since this does not correspond to the arc, its length

|

P

⁢

Q

¯

|

<

θ

.

Drop the

perpendicular

from

Q

to

O

⁢

P

¯

. Let

R

be the point of

intersection

. Note that

|

O

⁢

R

¯

|

=

cos

⁡

θ

and

|

Q

⁢

R

¯

|

=

sin

⁡

θ

.

Since

0

<

θ

<

π

2

,

0

<

sin

⁡

θ

<

1

and

0

<

cos

⁡

θ

<

1

. Thus,

R

≠

Q

and

R

lies strictly in between

O

and

P

. Therefore,

|

P

⁢

R

¯

|

>

0

.

By the

Pythagorean theorem

,

|

Q

⁢

R

¯

|

2

+

|

P

⁢

R

¯

|

2

=

|

P

⁢

Q

¯

|

2

. Thus,

|

Q

⁢

R

¯

|

2

<

|

P

⁢

Q

¯

|

2

. Therefore,

sin

⁡

θ

=

|

Q

⁢

R

¯

|

<

|

P

⁢

Q

¯

|

<

θ

.
∎

The analogous result for

θ

slightly below 0 is:

Corollary 1

.

Let

-

π

2

<

θ

<

0

, where

θ

is an angle measured in radians. Then

θ

<

sin

⁡

θ

.

Proof.

Since

0

<

-

θ

<

π

2

, the previous

theorem

yields

sin

⁡

(

-

θ

)

<

-

θ

. Since

sin

is an odd function,

-

sin

⁡

θ

<

-

θ

. It follows that

θ

<

sin

⁡

θ

.
∎

Title

comparison of

s

⁢

i

⁢

n

θ

and

θ

near

θ

=

0

Canonical name

ComparisonOfsinthetaAndthetaNeartheta0

Date of creation

2013-03-22 16:58:29

Last modified on

2013-03-22 16:58:29

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A03

Classification

msc 51N20

Classification

msc 26A06

Related topic

LimitOfDisplaystyleFracsinXxAsXApproaches0

Related topic

JordansInequality

Generated on Fri Feb 9 19:41:13 2018 by

LaTeXML
exponential

exponential

Preamble.

We use

ℝ

+

⊂

ℝ

to denote the set of
positive real numbers. Our aim is to define the

exponential

, or
the generalized power operation,

x

p

,

x

∈

ℝ

+

,

p

∈

ℝ

.

The power index

p

in the above
expression is called the

exponent

. We take it as proven that

ℝ

is a

complete

, ordered field. No other properties of the

real numbers

are invoked.

Definition.

For

x

∈

ℝ

+

and

n

∈

ℤ

we define

x

n

in terms of
repeated

multiplication

. To be more precise, we inductively
characterize

natural number

powers as follows:

x

0

=

1

,

x

n

+

1

=

x

⋅

x

n

,

n

∈

ℕ

.

The

existence

of the
reciprocal is guaranteed by the

assumption

that

ℝ

is a field.
Thus, for negative exponents, we can define

x

-

n

=

(

x

-

1

)

n

,

n

∈

ℕ

,

where

x

-

1

is the reciprocal of

x

.

The case of arbitrary exponents is somewhat more complicated. A
possible strategy is to define roots, then

rational

powers, and then
extend by

continuity

. Our approach is different. For

x

∈

ℝ

+

and

p

∈

ℝ

, we define the set of all reals that one would want
to be smaller than

x

p

, and then define the latter as the least

upper bound

of this set. To be more precise, let

x

>

1

and define

L

⁢

(

x

,

p

)

=

{

z

∈

ℝ

+

:

z

n

<

x

m

⁢

for all

⁢

m

∈

ℤ

,

n

∈

ℕ

⁢

such that

⁢

m

<

p

⁢

n

}

.

We then define

x

p

to be the least upper bound of

L

⁢

(

x

,

p

)

.
For

x

<

1

we define

x

p

=

(

x

-

1

)

-

p

.

The exponential operation possesses a number of
important properties (

http://

planetmath

.org/PropertiesOfTheExponential

),
some of which characterize it up to uniqueness.

Note.

It is also possible to define the exponential operation in
terms of the

exponential function

and the

natural logarithm

. Since these

concepts

require
the context of

differential

theory, it seems preferable to give a basic definition
that relies only on the foundational property of the reals.

Title

exponential

Canonical name

Exponential

Date of creation

2013-03-22 12:29:59

Last modified on

2013-03-22 12:29:59

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

17

Author

rmilson (146)

Entry type

Definition

Classification

msc 26A03

Synonym

exponential operation

Related topic

RealNumber

Defines

exponent

Defines

power

Generated on Fri Feb 9 19:41:18 2018 by

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fraction power

fraction power

Let

m

be an

integer

and

n

a

positive

factor of

m

.  If

x

is a positive real number, we may write the identical equation

(

x

m

n

)

n

=

x

m

n

⋅

n

=

x

m

and therefore the

definition

of

n

th

root (

http://planetmath.org/NthRoot

) gives the

x

m

n

=

x

m

n

.

(1)

Here, the

exponent

m

n

is an integer.  For enabling the validity of (1) for the cases where

n

does not divide

m

we must set the following

Definition.

Let

m

n

be a

fractional number

, i.e. an integer

m

not

divisible

by the integer

n

, which latter we assume to be positive.  For any positive real number

x

we define the

fraction power

x

m

n

as the

n

th

x

m

n

:=

x

m

n

.

(2)

Remarks

1.

The existence of the in the right
hand side of (2) is proved
here (

http://planetmath.org/existenceofnthroot

).

2.

The defining equation (2) is independent on the form of the exponent

m

n

:  If

k

l

=

m

n

,  then we have

(

x

m

n

)

l

⁢

n

=

[

(

x

m

n

)

n

]

l

=

x

l

⁢

m

=

x

k

⁢

n

=

[

(

x

k

l

)

l

]

n

=

(

x

k

l

)

l

⁢

n

,  and because the mapping

y

↦

y

l

⁢

n

is injective in

ℝ

+

, the positive numbers

x

k

l

and

x

m

n

must be equal.

3.

The

fraction power function

x

↦

x

m

n

is a special case of

power function

.

4.

The presumption that

x

is positive signifies that one can not identify all

n

th

roots (

http://planetmath.org/NthRoot

)

x

n

and the powers

x

1

n

.  For example,

-

8

3

equals

-

2

and

2

6

=

1

3

,  but one

must not

(

-

8

)

1

3

=

(

-

8

)

2

6

=

(

-

8

)

2

6

=

64

6

=

2

.

The point is that

(

-

8

)

1

3

is not defined in

ℝ

.  Here we have

l

=

6

and the mapping

y

↦

y

l

⁢

n

is not injective in

ℝ

-

∪

ℝ

+

.
— Nevertheless, some people and books may use also for negative

x

the equality

x

n

=

x

1

n

and more generally

x

m

n

=

x

m

n

where one then insists that

gcd

⁡

(

m

,

n

)

=

1

.

5.

According to the preceding item, for the negative values of

x

the

derivative

of odd roots (

http://planetmath.org/NthRoot

), e.g.

x

3

, ought to be calculated as follows:

d

⁢

x

3

d

⁢

x

=

d

⁢

(

-

-

x

3

)

d

⁢

x

=

-

d

⁢

(

-

x

)

1

3

d

⁢

x

=

-

1

3

⋅

(

-

x

)

-

2

3

⁢

(

-

1

)

=

1

3

⁢

(

-

x

)

2

3

=

1

3

⁢

x

2

3

The result is similar as

d

⁢

x

3

d

⁢

x

for positive

x

’s, although the root functions are not special cases of the power function.

Title

fraction power

Canonical

name

FractionPower

Date of creation

2014-09-21 12:12:39

Last modified on

2014-09-21 12:12:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

21

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A03

Synonym

fractional power

Related topic

PowerFunction

Related topic

GeneralPower

Related topic

IntegrationOfFractionPowerExpressions

Related topic

NthRootFormulas

Generated on Fri Feb 9 19:41:21 2018 by

LaTeXML
if d⁢(x_i,x_{i+1})<1/2^i then x_i is a Cauchy sequence

if

d

⁢

(

x

i

,

x

i

+

1

)

<

1

/

2

i

then

x

i

is a Cauchy sequence

Lemma 1

.

Suppose

x

1

,

x

2

,

…

, is a

sequence

in a

metric space

.
If for some

N

≥

1

, we have

d

⁢

(

a

i

,

a

i

+

1

)

<

1

/

2

i

for all

i

≥

N

,
then

{

x

i

}

is a

Cauchy sequence

.

Proof.

Let us denote by

d

the metric

function

.
If

ε

>

0

, then for some

N

∈

ℕ

we have

1

/

2

N

<

ε

.
Thus, if

N

<

m

<

n

we have

d

⁢

(

x

m

,

x

n

)

≤

d

⁢

(

x

m

,

x

m

+

1

)

+

⋯

+

d

⁢

(

x

n

-

1

,

x

n

)

=

(

1

2

)

m

+

⋯

+

(

1

2

)

n

-

1

=

(

1

2

)

m

-

1

⁢

∑

i

=

1

n

-

m

(

1

2

)

i

=

(

1

2

)

m

-

1

⁢

1

-

(

1

2

)

n

-

m

1

-

1

2

<

(

1

2

)

m

<

(

1

2

)

N

<

ε

,

where we have used the

triangle inequality

and the
geometric sum formula (

http://planetmath.org/GeometricSeries

).
∎

Title

if

d

⁢

(

x

i

,

x

i

+

1

)

<

1

/

2

i

then

x

i

is a Cauchy sequence

Canonical name

IfDxiXi112iThenXiIsACauchySequence

Date of creation

2013-03-22 14:37:31

Last modified on

2013-03-22 14:37:31

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Result

Classification

msc 26A03

Classification

msc 54E35

Generated on Fri Feb 9 19:41:23 2018 by

LaTeXML
interleave sequence

interleave sequence

Let

S

be a set, and let

{

x

i

}

,

i

=

0

,

1

,

2

,

…

and

{

y

i

}

,

i

=

0

,

1

,

2

,

…

be two

sequences

in

S

. The

interleave sequence

is defined to be the sequence

x

0

,

y

0

,

x

1

,

y

1

,

…

. Formally, it is the sequence

{

z

i

}

,

i

=

0

,

1

,

2

,

…

given by

z

i

:=

{

x

k

if

⁢

i

=

2

⁢

k

⁢

is even,

y

k

if

⁢

i

=

2

⁢

k

+

1

⁢

is odd.

Title

interleave sequence

Canonical name

InterleaveSequence

Date of creation

2013-03-22 11:52:12

Last modified on

2013-03-22 11:52:12

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 26A03

Classification

msc 40-00

Classification

msc 11A15

Generated on Fri Feb 9 19:41:25 2018 by

LaTeXML
limit inferior

limit inferior

Let

S

⊂

ℝ

be a set of real numbers. Recall that a

limit
point

of

S

is a

real number

x

∈

ℝ

such that for all

ϵ

>

0

there exist infinitely many

y

∈

S

such that

|

x

-

y

|

<

ϵ

.

We define

lim inf

⁡

S

, pronounced the

limit inferior

of

S

, to be the

infimum

of all the limit
points of

S

. If there are no limit points, we define the limit
inferior to be

+

∞

.

The two most common notations for the limit inferior are

lim inf

⁡

S

and

lim

¯

⁢

S

.

An alternative, but

equivalent

, definition is available in the case of
an

infinite

sequence

of real numbers

x

0

,

x

1

,

x

2

,

,

…

. For
each

k

∈

ℕ

, let

y

k

be the infimum of the

k

th

tail,

y

k

=

inf

j

≥

k

⁡

x

j

.

This construction produces a
non-decreasing sequence

y

0

≤

y

1

≤

y

2

≤

…

,

which either

converges

to its

supremum

, or diverges to

+

∞

.
We define the limit inferior of the original sequence to be this limit;

lim inf

k

⁡

x

k

=

lim

k

⁡

y

k

.

Title

limit inferior

Canonical name

LimitInferior

Date of creation

2013-03-22 12:22:01

Last modified on

2013-03-22 12:22:01

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 26A03

Synonym

liminf

Synonym

infimum limit

Related topic

LimitSuperior

Generated on Fri Feb 9 19:41:27 2018 by

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