aarjaneiro/planetmath huggingface dataset:
limit superior

limit superior

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 sup

⁡

S

=

lim

¯

, pronounced the

limit superior

of

S

, to be the

supremum

of all the limit
points of

S

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

-

∞

.

We can generalize the above definition to the case of a

mapping

f

:

X

→

ℝ

. Now, we define a limit point of

f

to be an

x

∈

ℝ

such that for all

ϵ

>

0

there exist infinitely many

y

∈

X

such that

|

x

-

f

⁢

(

y

)

|

<

ϵ

.

We then define

lim sup

⁡

f

, to be the
supremum of all the limit points of

f

, or

-

∞

if there are no
limit points. We recover the previous definition as a special case by
considering the limit superior of the

inclusion mapping

ι

:

S

→

ℝ

.

Since a

sequence

of real numbers

x

0

,

x

1

,

x

2

,

,

…

is just a
mapping from

ℕ

to

ℝ

, we may adapt the above definition
to arrive at the notion of the limit superior of a sequence. However
for the case of sequences, an alternative, but

equivalent

definition
is available. For each

k

∈

ℕ

, let

y

k

be the supremum of
the

k

th

tail,

y

k

=

sup

j

≥

k

⁡

x

j

.

This construction produces a
non-increasing sequence

y

0

≥

y

1

≥

y

2

≥

…

,

which either

converges

to its

infimum

, or diverges to

-

∞

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

lim sup

k

⁡

x

k

=

lim

k

⁡

y

k

.

Title

limit superior

Canonical name

LimitSuperior

Date of creation

2013-03-22 12:21:58

Last modified on

2013-03-22 12:21:58

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

12

Author

rmilson (146)

Entry type

Definition

Classification

msc 26A03

Synonym

limsup

Synonym

supremum limit

Related topic

LimitInferior

Generated on Fri Feb 9 19:41:29 2018 by

LaTeXML
open and closed intervals have the same cardinality

open and closed intervals have the same cardinality

Proposition

.

The sets of real numbers

[

0

,

1

]

,

[

0

,

1

)

,

(

0

,

1

]

, and

(

0

,

1

)

all have the same

cardinality

.

We give two proofs of this

proposition

.

Proof.

Define a map

f

:

[

0

,

1

]

→

[

0

,

1

]

by

f

⁢

(

x

)

=

(

x

+

1

)

/

3

. The map

f

is

strictly increasing

, hence

injective

. Moreover, the image of

f

is

contained

in the

interval

[

1

3

,

2

3

]

⊊

(

0

,

1

)

, so the maps

f

r

:

[

0

,

1

]

→

[

0

,

1

)

and

f

o

:

[

0

,

1

]

→

(

0

,

1

)

obtained from

f

by restricting the

codomain

are both injective. Since the

inclusions

into

[

0

,

1

]

are also injective, the Cantor-Schröder-Bernstein

theorem

(

http://planetmath.org/SchroederBernsteinTheorem

) can be used to construct

bijections

h

r

:

[

0

,

1

]

→

[

0

,

1

)

and

h

o

:

[

0

,

1

]

→

(

0

,

1

)

. Finally, the map

r

:

(

0

,

1

]

→

[

0

,

1

)

defined by

r

⁢

(

x

)

=

1

-

x

is a bijection.

Since having the same cardinality is an

equivalence relation

, all four intervals have the same cardinality.
∎

Proof.

Since

[

0

,

1

]

∩

ℚ

is

countable

, there is a bijection

a

:

ℕ

→

[

0

,

1

]

∩

ℚ

. We may select

a

so that

a

⁢

(

0

)

=

0

and

a

⁢

(

1

)

=

1

. The map

f

:

[

0

,

1

]

∩

ℚ

→

(

0

,

1

)

∩

ℚ

defined by

f

⁢

(

x

)

=

a

⁢

(

a

-

1

⁢

(

x

)

+

2

)

is a bijection because it is a composition of bijections. A bijection

h

:

[

0

,

1

]

→

(

0

,

1

)

can be constructed by gluing the map

f

to the

identity map

on

(

0

,

1

)

∖

ℚ

. The

formula

for

h

is

h

⁢

(

x

)

=

{

f

⁢

(

x

)

,

x

∈

ℚ

x

,

x

∉

ℚ

.

The other bijections can be constructed similarly.
∎

The reasoning above can be extended to show that any two arbitrary intervals in

ℝ

have the same cardinality.

Title

open and

closed intervals

have the same cardinality

Canonical name

OpenAndClosedIntervalsHaveTheSameCardinality

Date of creation

2013-03-22 15:43:32

Last modified on

2013-03-22 15:43:32

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Result

Classification

msc 26A03

Classification

msc 03E10

Generated on Fri Feb 9 19:41:31 2018 by

LaTeXML
power rule

power rule

The

power rule

states that

d

d

⁢

x

⁢

x

p

=

p

⁢

x

p

-

1

,

p

∈

ℝ

This rule, when combined with the

chain rule

,

product rule

, and

sum rule

,
makes calculating many

derivatives

far more tractable. This rule can be derived by repeated application of the product rule.
See the proof of the power rule (

http://planetmath.org/ProofOfPowerRule

).

Repeated use of the above formula gives

d

i

d

⁢

x

i

⁢

x

k

=

{

0

i

>

k

k

!

(

k

-

i

)

!

⁢

x

k

-

i

i

≤

k

,

for

i

,

k

∈

ℤ

.

Examples

d

d

⁢

x

⁢

x

0

=

0

x

=

0

=

d

d

⁢

x

⁢

1

d

d

⁢

x

⁢

x

1

=

1

⁢

x

0

=

1

=

d

d

⁢

x

⁢

x

d

d

⁢

x

⁢

x

2

=

2

⁢

x

d

d

⁢

x

⁢

x

3

=

3

⁢

x

2

d

d

⁢

x

⁢

x

=

d

d

⁢

x

⁢

x

1

/

2

=

1

2

⁢

x

-

1

/

2

=

1

2

⁢

x

d

d

⁢

x

⁢

2

⁢

x

e

=

2

⁢

e

⁢

x

e

-

1

Title

power rule

Canonical name

PowerRule

Date of creation

2013-03-22 12:28:03

Last modified on

2013-03-22 12:28:03

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 26A03

Related topic

ProductRule

Related topic

Derivation

Related topic

Derivative

Generated on Fri Feb 9 19:41:33 2018 by

LaTeXML
properties of the exponential

properties of the exponential

The

exponential operation

possesses the following properties.

•

For

x

,

y

∈

ℝ

+

,

p

∈

ℝ

we have

(

x

⁢

y

)

p

=

x

p

⁢

y

p

•

For

x

∈

ℝ

+

we have

x

0

=

1

,

x

1

=

x

,

x

p

+

q

=

x

p

⁢

x

q

,

(

x

p

)

q

=

x

p

⁢

q

p

,

q

∈

ℝ

.

•

Monotonicity. (

http://planetmath.org/TotalOrder

) For

x

,

y

∈

R

+

with

x

<

y

and

p

∈

R

+

we have

x

p

<

y

p

,

x

-

p

>

y

-

p

.

•

Continuity

.
The exponential operation is

continuous

with respect to its

arguments

. To be more precise, the following

function

is continuous:

P

:

ℝ

+

×

ℝ

→

ℝ

,

P

⁢

(

x

,

y

)

=

x

y

.

Let us also note that the exponential operation is characterized (in
the sense of existence and uniqueness) by the

additivity

and

continuity

properties. [

Author’s note

: One can probably get away with
substantially less, but I haven’t given this enough thought.]

Title

properties of the

exponential

Canonical name

PropertiesOfTheExponential

Date of creation

2013-03-22 12:30:02

Last modified on

2013-03-22 12:30:02

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

15

Author

rmilson (146)

Entry type

Theorem

Classification

msc 26A03

Generated on Fri Feb 9 19:41:40 2018 by

LaTeXML
squeeze rule

squeeze rule

Squeeze rule

for

sequences

Let

f

,

g

,

h

:

ℕ

→

ℝ

be three sequences of

real numbers

such that

f

⁢

(

n

)

≤

g

⁢

(

n

)

≤

h

⁢

(

n

)

for all

n

. If

lim

n

→

∞

⁡

f

⁢

(

n

)

and

lim

n

→

∞

⁡

h

⁢

(

n

)

exist
and are equal, say to

a

, then

lim

n

→

∞

⁡

g

⁢

(

n

)

also exists and
equals

a

.

The proof is fairly straightforward. Let

ϵ

be any real number

>

0

.
By

hypothesis

there exist

M

,

N

∈

ℕ

such that

|

a

-

f

⁢

(

n

)

|

<

ϵ

⁢

for all

⁢

n

≥

M

|

a

-

h

⁢

(

n

)

|

<

ϵ

⁢

for all

⁢

n

≥

N

Write

L

=

max

⁡

(

M

,

N

)

. For

n

≥

L

we have

•

if

g

⁢

(

n

)

≥

a

:

|

g

⁢

(

n

)

-

a

|

=

g

⁢

(

n

)

-

a

≤

h

⁢

(

n

)

-

a

<

ϵ

•

else

g

⁢

(

n

)

<

a

and:

|

g

⁢

(

n

)

-

a

|

=

a

-

g

⁢

(

n

)

≤

a

-

f

⁢

(

n

)

<

ϵ

So, for all

n

≥

L

, we have

|

g

⁢

(

n

)

-

a

|

<

ϵ

, which is the desired

conclusion

.

Squeeze rule for

functions

Let

f

,

g

,

h

:

S

→

ℝ

be three real-valued functions on a neighbourhood

S

of a real number

b

, such that

f

⁢

(

x

)

≤

g

⁢

(

x

)

≤

h

⁢

(

x

)

for all

x

∈

S

-

{

b

}

. If

lim

x

→

b

⁡

f

⁢

(

x

)

and

lim

x

→

b

⁡

h

⁢

(

x

)

exist and are equal, say to

a

, then

lim

x

→

b

⁡

g

⁢

(

x

)

also exists
and equals

a

.

Again let

ϵ

be an arbitrary positive real number. Find

positive

reals

α

and

β

such that

|

a

-

f

⁢

(

x

)

|

<

ϵ

⁢

whenever

⁢

0

<

|

b

-

x

|

<

α

|

a

-

h

⁢

(

x

)

|

<

ϵ

⁢

whenever

⁢

0

<

|

b

-

x

|

<

β

Write

δ

=

min

⁡

(

α

,

β

)

. Now, for any

x

such that

|

b

-

x

|

<

δ

, we have

•

if

g

⁢

(

x

)

≥

a

:

|

g

⁢

(

x

)

-

a

|

=

g

⁢

(

x

)

-

a

≤

h

⁢

(

x

)

-

a

<

ϵ

•

else

g

⁢

(

x

)

<

a

and:

|

g

⁢

(

x

)

-

a

|

=

a

-

g

⁢

(

x

)

≤

a

-

f

⁢

(

x

)

<

ϵ

and we are done.

Title

squeeze rule

Canonical

name

SqueezeRule

Date of creation

2013-03-22 13:46:31

Last modified on

2013-03-22 13:46:31

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

4

Author

Daume (40)

Entry type

Theorem

Classification

msc 26A03

Synonym

squeeze theorem

Synonym

squeeze test

Generated on Fri Feb 9 19:41:43 2018 by

LaTeXML
all derivatives of sinc are bounded by 1

all derivatives of sinc are bounded by

1

Let us show that all

derivatives

of

sinc

are bounded by

1

.

First of all, let us out that

sinc

⁡

(

t

)

≤

1

is
bounded by the Jordan’s

inequality

. To the derivatives, let
us write

sinc

as a

Fourier integral

,

sinc

⁡

(

t

)

=

1

2

⁢

∫

-

1

1

e

i

⁢

x

⁢

t

⁢

𝑑

x

.

Let

k

=

1

,

2

,

…

. Then

d

k

d

⁢

t

k

⁢

sinc

⁡

(

t

)

=

1

2

⁢

∫

-

1

1

(

i

⁢

x

)

k

⁢

e

i

⁢

x

⁢

t

⁢

𝑑

x

.

and

|

d

k

d

⁢

t

k

⁢

sinc

⁡

(

t

)

|

=

|

1

2

⁢

∫

-

1

1

(

i

⁢

x

)

k

⁢

e

i

⁢

x

⁢

t

⁢

𝑑

x

|

≤

1

2

⁢

∫

-

1

1

|

(

i

⁢

x

)

k

⁢

e

i

⁢

x

⁢

t

|

⁢

𝑑

x

≤

1

2

⁢

∫

-

1

1

|

x

|

k

⁢

𝑑

x

≤

1

2

⋅

2

⁢

∫

0

1

|

x

|

k

⁢

𝑑

x

≤

∫

0

1

x

k

⁢

𝑑

x

≤

1

k

+

1

<

1

.

Title

all derivatives of sinc are bounded by

1

Canonical name

AllDerivativesOfSincAreBoundedBy1

Date of creation

2013-03-22 15:39:03

Last modified on

2013-03-22 15:39:03

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

10

Author

matte (1858)

Entry type

Result

Classification

msc 26A06

Generated on Fri Feb 9 19:41:45 2018 by

LaTeXML
alternate statement of Bolzano-Weierstrass theorem

alternate statement of Bolzano-Weierstrass theorem

Theorem

.

Every

bounded

,

infinite set

of

real numbers

has a

limit point

.

Proof.

Let

S

⊂

ℝ

be bounded and

infinite

. Since

S

is bounded there exist

a

,

b

∈

ℝ

, with

a

<

b

, such that

S

⊂

[

a

,

b

]

. Let

b

-

a

=

l

and denote the

midpoint

of the

interval

[

a

,

b

]

by

m

. Note that at least one of

[

a

,

m

]

,

[

m

,

b

]

must contain infinitely many points of

S

; select an interval satisfying this condition, denoting its left

endpoint

by

a

1

and its right endpoint by

b

1

. Continuing this process inductively, for each

n

∈

ℕ

, we have an interval

[

a

n

,

b

n

]

satisfying

[

a

n

,

b

n

]

⊂

[

a

n

-

1

,

b

n

-

1

]

⊂

⋯

⊂

[

a

1

,

b

1

]

⊂

[

a

,

b

]

⁢

,

(1)

where, for each

i

∈

ℕ

such that

1

≤

i

≤

n

, the interval

[

a

i

,

b

i

]

contains infinitely many points of

S

and is of length

l

/

2

i

. Next we note that the set

A

=

{

a

1

,

a

2

⁢

…

,

a

n

}

is

contained

in

[

a

,

b

]

, hence is bounded, and as such, has a

supremum

which we denote by

x

. Now, given

ϵ

>

0

, there exists

N

∈

ℕ

such that

x

-

ϵ

<

a

N

≤

x

. Furthermore, for every

m

≥

N

, we have

x

-

ϵ

<

a

N

≤

a

m

≤

x

. In particular, if we select

m

≥

N

such that

l

/

2

m

<

ϵ

, then we have

x

-

ϵ

<

a

n

≤

a

m

≤

x

≤

b

m

=

a

m

+

l

2

m

<

x

+

ϵ

⁢

.

(2)

Since

[

a

m

,

b

m

]

⊂

(

x

-

ϵ

,

x

+

ϵ

)

contains infinitely many points of

S

, we may conclude that

x

is a limit point of

S

.
∎

Title

alternate statement of Bolzano-Weierstrass theorem

Canonical name

AlternateStatementOfBolzanoWeierstrassTheorem

Date of creation

2013-03-22 16:40:13

Last modified on

2013-03-22 16:40:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 26A06

Related topic

BolzanoWeierstrassTheorem

Related topic

LimitPoint

Related topic

Bounded

Related topic

Infinite

Generated on Fri Feb 9 19:41:47 2018 by

LaTeXML
alternating harmonic series

alternating harmonic series

The

alternating harmonic series

is given by the following

infinite series

:

∑

i

=

1

∞

(

-

1

)

n

+

1

n

.

(1)

The series

converges

to

ln

⁡

2

and it is the prototypical example of a

conditionally convergent

series.

First, notice that the series is not absolutely convergent. By taking the

absolute value

of each term, we get the

harmonic series

, which is divergent. There are several ways to show this, and we invite the reader to the entry on harmonic series for further exploration.

Next, to show that the series (1) converges, we use the

alternating series test

(

http://planetmath.org/AlternatingSeriesTest

): since

lim

n

→

∞

⁡

1

n

=

0

,

the

alternating series

(1) converges.

Remarks

.

•

Other examples of conditionally convergent series can be discovered using variants of the alternating harmonic series. For instance, the following series

∑

i

=

1

∞

(

-

1

)

n

⁢

n

2

+

cos

⁡

n

n

3

-

n

2

+

e

-

n

.

can easily be shown to be conditionally convergent. Here is another example, more of a generalization, called the :

∑

i

=

1

∞

(

-

1

)

n

+

1

n

p

,

(2)

where

p

is non-negative

real number

. The

convergence

of the is tabulated below:

p

convergence

(

1

,

∞

)

absolutely convergent

(

0

,

1

]

conditionally convergent

0

divergent

•

U

s

i

n

g

R

i

e

m

a

n

n

s

e

r

i

e

s

t

h

e

o

r

e

m

,

o

n

e

e

a

s

i

l

y

s

e

e

s

t

h

a

t

n

o

t

e

v

e

r

y

c

o

n

d

i

t

i

o

n

a

l

l

y

c

o

n

v

e

r

g

e

n

t

s

e

r

i

e

s

i

s

a

l

t

e

r

n

a

t

i

n

g

.

B

y

a

p

p

r

o

p

r

i

a

t

e

l

y

r

e

a

r

r

a

n

g

i

n

g

t

h

e

a

l

t

e

r

n

a

t

i

n

g

h

a

r

m

o

n

i

c

s

e

r

i

e

s

,

o

n

e

g

e

t

s

a

c

o

n

d

i

t

i

o

n

a

l

l

y

c

o

n

v

e

r

g

e

n

t

s

e

r

i

e

s

t

h

a

t

i

s

n

o

t

a

l

t

e

r

n

a

t

i

n

g

:

d

e

f

i

n

e

σ:

N→N

a

⁢

s

⁢

f

⁢

o

⁢

l

⁢

l

⁢

o

⁢

w

⁢

s

:

:=

⁢
σ

(
i
)

{

+

⁢
2
i

1

3

≡

⁢
if
2
i

-
1

(

mod
3

)

-

⁢
4
i

2

3

≡

⁢
if
2
i

1

(

mod
3

)

⁢
4
i

3

otherwise

\@unrecurse

Generated on Fri Feb 9 19:41:57 2018 by

LaTeXML
antipodal isothermic points

antipodal isothermic points

Assume that the momentary temperature on any

great circle

of a sphere varies continuously (

http://planetmath.org/

Continuous

).  Then there exist two

diametral points

(i.e.

antipodal points

,

end points

of a certain diametre (

http://planetmath.org/

Diameter

)) having the same temperature.

Proof.

Denote by

x

the

distance

of any point

P

measured in a certain direction along the great circle from a and let

T

⁢

(

x

)

be the temperature in

P

.  Then we have a continuous (and periodic (

http://planetmath.org/PeriodicFunctions

)) real function

T

defined for

x

≧

0

satisfying

T

⁢

(

x

+

p

)

=

T

⁢

(

x

)

where

p

is the perimetre of the circle.  Then also the

function

f

defined by

f

⁢

(

x

)

:=

T

⁢

(

x

+

p

2

)

-

T

⁢

(

x

)

,

i.e. the temperature

difference

in two antipodic (diametral) points of the great circle, is continuous.  We have

f

⁢

(

p

2

)

=

T

⁢

(

p

)

-

T

⁢

(

p

2

)

=

T

⁢

(

0

)

-

T

⁢

(

p

2

)

=

-

f

⁢

(

0

)

.

(1)

If

f

happens to vanish in

x

=

0

,  then the temperature is the same in

x

=

p

2

and the assertion proved.  But if

f

⁢

(

0

)

≠

0

,  then by (1), the values of

f

in

x

=

0

and in

x

=

p

2

have opposite signs.  Therefore, by Bolzano’s

theorem

, there exists a point

ξ

between

0

and

p

2

such that

f

⁢

(

ξ

)

=

0

.  Thus the temperatures in

ξ

and

ξ

+

π

2

are the same.

Reference:

http://www.maths.lth.se/query/

Fråga Lund om matematik, 6 april 2006

Title

antipodal isothermic points

Canonical name

AntipodalIsothermicPoints

Date of creation

2013-03-22 18:32:10

Last modified on

2013-03-22 18:32:10

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Application

Classification

msc 26A06

Generated on Fri Feb 9 19:41:59 2018 by

LaTeXML
application of fundamental theorem of integral calculus

application of fundamental theorem of integral calculus

We will derive the

addition formulas

of the sine and the

cosine functions

supposing known only their

derivatives

and the

chain rule

.

Define the

function

F

:

ℝ

→

ℝ

through

F

⁢

(

x

)

:=

[

sin

⁡

x

⁢

cos

⁡

α

+

cos

⁡

x

⁢

sin

⁡

α

-

sin

⁡

(

x

+

α

)

]

2

+

[

cos

⁡

x

⁢

cos

⁡

α

-

sin

⁡

x

⁢

sin

⁡

α

-

cos

⁡

(

x

+

α

)

]

2

where

α

is, for the , a constant.  The derivative of

F

is easily calculated:

F

′

⁢

(

x

)

=

⁢

2

⁢

[

sin

⁡

x

⁢

cos

⁡

α

+

cos

⁡

x

⁢

sin

⁡

α

-

sin

⁡

(

x

+

α

)

]

⁢

[

cos

⁡

x

⁢

cos

⁡

α

-

sin

⁡

x

⁢

sin

⁡

α

-

cos

⁡

(

x

+

α

)

]

+

2

⁢

[

cos

⁡

x

⁢

cos

⁡

α

-

sin

⁡

x

⁢

sin

⁡

α

-

cos

⁡

(

x

+

α

)

]

⁢

[

-

sin

⁡

x

⁢

cos

⁡

α

-

cos

⁡

x

⁢

sin

⁡

α

+

sin

⁡

(

x

+

α

)

]

But this expression is identically 0.  By the

fundamental theorem of integral calculus

,

F

must be a

constant function

.  Since

F

⁢

(

0

)

=

0

,  we have

F

⁢

(

x

)

≡

0

for any

x

and naturally also for any

α

.  Because

F

⁢

(

x

)

is a sum of two squares, the both addends of it have to vanish identically, which yields the equalities

sin

⁡

x

⁢

cos

⁡

α

+

cos

⁡

x

⁢

sin

⁡

α

-

sin

⁡

(

x

+

α

)

=

0

,

cos

⁡

x

⁢

cos

⁡

α

-

sin

⁡

x

⁢

sin

⁡

α

-

cos

⁡

(

x

+

α

)

=

0

.

These the addition formulas (

http://planetmath.org/GoniometricFormulae

)

sin

⁡

(

x

+

α

)

=

sin

⁡

x

⁢

cos

⁡

α

+

cos

⁡

x

⁢

sin

⁡

α

,

cos

⁡

(

x

+

α

)

=

cos

⁡

x

⁢

cos

⁡

α

-

sin

⁡

x

⁢

sin

⁡

α

.

Title

application of fundamental theorem of integral calculus

Canonical name

ApplicationOfFundamentalTheoremOfIntegralCalculus

Date of creation

2013-03-22 18:50:52

Last modified on

2013-03-22 18:50:52

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Example

Classification

msc 26A06

Related topic

TrigonometricFormulasFromSeries

Generated on Fri Feb 9 19:42:01 2018 by

LaTeXML
Barbălat’s lemma

Barbălat’s lemma

Lemma

(Barbălat).

Let

f

:

(

0

,

∞

)

→

R

be

Riemann integrable

and

uniformly continuous

then

lim

t

→

∞

⁡

f

⁢

(

t

)

=

0

.

Note that if

f

is non-negative, then Riemann integrability is the same as being

L

1

in the sense of Lebesgue, but if

f

oscillates then the

Lebesgue integral

may not exist.

Further note that the uniform continuity is required to prevent sharp “spikes” that might prevent the limit from existing. For example suppose we add a spike of height 1 and area

2

-

n

at every integer. Then the function is

continuous

and

L

1

(and thus Riemann integrable), but

f

⁢

(

t

)

would not have a

limit at infinity

.

References

1

Hartmut Logemann, Eugene P. Ryan.
.

The American Mathematical Monthly

, 111(10):864–889,
2004.

Title

Barbălat’s lemma

Canonical name

BarbualatsLemma

Date of creation

2013-03-22 14:52:31

Last modified on

2013-03-22 14:52:31

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 26A06

Synonym

Barbalat’s lemma

Generated on Fri Feb 9 19:42:03 2018 by

LaTeXML
binomial formula

binomial formula

The

binomial formula

gives the

power series

expansion of the

p

th

power function

. The power

p

can be an

integer

,

rational

, real, or even a

complex number

. The

formula

is

(

1

+

x

)

p

=

∑

n

=

0

∞

p

n

¯

n

!

⁢

x

n

=

∑

n

=

0

∞

(

p

n

)

⁢

x

n

where

p

n

¯

=

p

⁢

(

p

-

1

)

⁢

…

⁢

(

p

-

n

+

1

)

denotes the

falling
factorial

, and where

(

p

n

)

denotes the

generalized binomial
coefficient

.

For

p

=

0

,

1

,

2

,

…

the power series reduces to a

polynomial

, and we
obtain the usual

binomial theorem

. For other values of

p

, the

radius of convergence

of the series is

1

; the right-hand series

converges

pointwise

for all

complex

|

x

|

<

1

to the value on the left
side. Also note that the binomial formula is valid at

x

=

±

1

, but
for certain values of

p

only. Of course, we have

convergence

if

p

is a

natural number

. Furthermore, for

x

=

1

and real

p

, we have

absolute convergence

if

p

>

0

, and

conditional convergence

if

-

1

<

p

<

0

. For

x

=

-

1

we have absolute convergence for

p

>

0

.

Title

binomial formula

Canonical name

BinomialFormula

Date of creation

2013-03-22 12:23:52

Last modified on

2013-03-22 12:23:52

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

11

Author

rmilson (146)

Entry type

Theorem

Classification

msc 26A06

Synonym

Newton’s

binomial series

Related topic

BinomialTheorem

Related topic

GeneralizedBinomialCoefficients

Generated on Fri Feb 9 19:42:05 2018 by

LaTeXML