aarjaneiro/planetmath huggingface dataset:
uncountable sums of positive numbers

uncountable sums of positive numbers

The notion of sum of a series can be generalized to sums of nonnegative
real numbers over arbitrary

index sets

.

let

I

be a set and let

c

be a

mapping

from

I

to the nonnegative
real numbers. Then we may define the sum as follows:

∑

i

∈

I

c

i

=

sup

s

⊂

I

#

⁢

s

<

∞

⁡

∑

i

∈

s

c

i

In words, we are taking the

supremum

over all sums over finite subsets
of the index set. This agrees with the usual notion of sum when our
set is

countably infinite

, but generalizes this notion to

uncountable

index sets.

An important fact about this

generalization

is that the sum can only
be finite if the number of

elements

i

∈

I

such that

c

i

>

0

is

countable

. To demonstrate this fact, define the sets

s

n

(where n
is a nonnegative integer) as follows:

s

0

=

{

i

∈

I

∣

c

i

≥

1

}

when

n

>

0

,

s

n

=

{

i

∈

I

∣

1

/

n

>

c

i

≥

1

/

(

n

+

1

)

}

If any of these sets is

infinite

, then the sum will

diverge

so, for
the sum to be finite, all these sets must be finite. However, if
these sets are all finite, then their union is countable. In other
words, the number of indices for which

c

i

>

0

will be countable.

This notion finds use in places such as non-separable

Hilbert spaces

.
For

instance

, given a vector in such a space and a

complete

orthonormal set, one can express the norm of the vector as the sum of
the squares of its components using this definition even when the
orthonormal set is

uncountably infinite

.

This discussion can also be phrased in terms of Lesbegue integration
with respect to

counting measure

. For this point of view, please see the entry

support of integrable function with respect to counting measure is countable

.

Title

uncountable sums of positive numbers

Canonical name

UncountableSumsOfPositiveNumbers

Date of creation

2013-03-22 15:44:47

Last modified on

2013-03-22 15:44:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 40-00

Related topic

SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable

Generated on Fri Feb 9 21:10:59 2018 by

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(1+α/n)^n is monotone for large n

(

1

+

α

n

)

n

is monotone for large

n

Lemma

.

Let

α

be a

real number

. The sequence

(

1

+

α

n

)

n

is

monotone increasing

for all

n

>

|

α

|

.

Proof.

Let

n

>

|

α

|

. We want to prove the following

inequality

:

(

1

+

α

n

)

n

≤

(

1

+

α

n

+

1

)

n

+

1

Since both sides are

positive

, this follows by taking the

(

n

+

1

)

-th root and using the

arithmetic-geometric-harmonic means
inequality

:

(

1

+

α

n

)

n

n

+

1

=

1

⋅

(

1

+

α

n

)

⁢

⋯

⁢

(

1

+

α

n

)

n

+

1

⏟

n

+

1

elements

≤

1

+

n

⁢

(

1

+

α

n

)

n

+

1

=

1

+

α

n

+

1

∎

Title

(

1

+

α

n

)

n

is

monotone

for large

n

Canonical

name

1fracalphannIsMonotoneForLargeN

Date of creation

2013-03-22 17:53:55

Last modified on

2013-03-22 17:53:55

Owner

uriw (288)

Last modified by

uriw (288)

Numerical id

4

Author

uriw (288)

Entry type

Theorem

Classification

msc 40-01

Classification

msc 00-01

Generated on Fri Feb 9 21:11:01 2018 by

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3-manifold

3-manifold

3-manifold

A 3 dimensional

manifold

is a

topological space

which is

locally homeomorphic

to the

euclidean space

𝐑

3

.

One can see from simple constructions the great

variety

of objects that indicate that they are worth to study.

First without

boundary

:

1.

For example, with the

cartesian product

we can get:

–

S

2

×

S

1

–

𝐑

⁢

P

2

×

S

1

–

T

×

S

1

–

…

2.

Also by the

generalization

of the cartesian product:

fiber bundles

, one can build bundles

E

of the type

F

⊂

E

→

S

1

where

F

is any closed

surface

.

3.

Or interchanging the roles, bundles as:

S

1

⊂

E

→

F

For the second type it is known that for each

isotopy

class

[

ϕ

]

of maps

F

→

F

correspond to an unique bundle

E

ϕ

. Any

homeomorphism

f

:

F

→

F

representing the isotopy class

[

ϕ

]

is called a

monodromy

for

E

ϕ

.

From the previuos paragraph we infer that the

mapping class group

play a important role inthe understanding at least for this

subclass

of objets.

For the third class above one can use an

orbifold

instead of a simple surface to get a class of 3-manifolds called

Seifert fiber spaces

which are a large class of spaces needed to understand the modern classifications for 3-manifolds.

References

[G

] J.C. Gómez-Larrañaga.

3-manifolds which are unions of three solid tori

,
Manuscripta Math. 59 (1987), 325-330.

[GGH

] J.C. Gómez-Larrañaga, F.J. González-Acuña, J. Hoste.

Minimal

Atlases on 3-manifolds

,
Math. Proc. Camb. Phil. Soc. 109 (1991), 105-115.

[H

] J. Hempel.

3-manifolds

, Princeton University Press 1976.

[O

] P. Orlik.

Seifert Manifolds

, Lecture Notes in Math. 291,
1972 Springer-Verlag.

[S

] P. Scott.

The geometry of 3-manifolds

, Bull. London Math. Soc. 15 (1983), 401-487.

Title

3-manifold

Canonical name

3manifold

Date of creation

2013-03-11 19:23:28

Last modified on

2013-03-11 19:23:28

Owner

juanman (12619)

Last modified by

(0)

Numerical id

1

Author

juanman (0)

Entry type

Definition

Generated on Fri Feb 9 21:11:03 2018 by

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a pathological function of Riemann

a pathological function of Riemann

The

periodic

mantissa function

t

↦

t

-

⌊

t

⌋

has at each integer value of

t

a jump (saltus) equal to

-

1

, being in these points

continuous

from the right but not from the left.  For every real value

t

, one has

0

≦

t

-

⌊

t

⌋

<

1

.

(1)

Let us consider the series

∑

n

=

1

∞

n

⁢

x

-

⌊

n

⁢

x

⌋

n

2

(2)

due to Riemann.  Since by (1), all values of

x

∈

ℝ

and

n

∈

ℤ

+

satisfy

0

≦

n

⁢

x

-

⌊

n

⁢

x

⌋

n

2

<

1

n

2

,

(3)

the series is, by Weierstrass’ M-test,

uniformly convergent

on the whole

ℝ

(see also the p-test).  We denote by

S

⁢

(

x

)

the sum

function

of (2).

The

n

th

term of the series (2) defines a

periodic function

x

↦

n

⁢

x

-

⌊

n

⁢

x

⌋

n

2

(4)

with the period (

http://planetmath.org/PeriodicFunctions

)

1

n

and having especially for

0

≦

x

<

1

n

the value

x

n

.  The only points of

discontinuity

of this function are

x

=

m

n

(

m

=

0

,

±

1

,

±

2

,

…

)

,

(5)

where it

vanishes

and where it is continuous from the right but not from the left; in the point (5) this function apparently has the jump

-

1

n

2

.

The theorem of the entry

one-sided continuity by series

implies that the sum function

S

⁢

(

x

)

is continuous in every irrational point

x

, because the series (2) is uniformly convergent for every

x

and its terms are continuous for irrational points

x

.

Since the terms (4) of (2) are continuous from the right in the rational points (5), the same theorem implies that

S

⁢

(

x

)

is in these points continuous from the right.  It can be shown that

S

⁢

(

x

)

is in these points

discontinuous

from the left having the jump equal to

-

π

2

6

⁢

n

2

.

References

1

E. Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset III.2

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

Title

a

pathological function of Riemann

Canonical name

APathologicalFunctionOfRiemann

Date of creation

2013-03-22 18:34:17

Last modified on

2013-03-22 18:34:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Example

Classification

msc 40A05

Classification

msc 26A15

Classification

msc 26A03

Synonym

example of semicontinuous function

Related topic

DirichletsFunction

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Generated on Fri Feb 9 21:11:14 2018 by

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a series related to harmonic series

a series related to harmonic series

The series

∑

n

=

1

∞

1

n

⁢

n

n

=

∑

n

=

1

∞

1

n

1

+

1

n

(1)

is

divergent

.  In fact, since for every

positive integer

n, one has

2

n

>

n

,  i.e.

n

n

<

2

, any of the series satisfies

1

n

⁢

n

n

>

1

2

⁢

n

.

Because the

harmonic series

and therefore also

∑

1

∞

1

2

⁢

n

diverges

, the

comparison test

implies that the series (1) diverges.

Title

a

series related to harmonic series

Canonical name

ASeriesRelatedToHarmonicSeries

Date of creation

2013-03-22 17:56:40

Last modified on

2013-03-22 17:56:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Example

Classification

msc 40A05

Related topic

PTest

Related topic

RaabesCriteria

Generated on Fri Feb 9 21:11:16 2018 by

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Abel’s lemma

Abel’s lemma

Theorem

1

Let

{

a

i

}

i

=

0

N

and

{

b

i

}

i

=

0

N

be

sequences

of
real (or complex) numbers with

N

≥

0

.
For

n

=

0

,

…

,

N

, let

A

n

be the

partial sum

A

n

=

∑

i

=

0

n

a

i

.
Then

∑

i

=

0

N

a

i

⁢

b

i

=

∑

i

=

0

N

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

N

⁢

b

N

.

In the

trivial

case, when

N

=

0

, then sum on the right hand side
should be interpreted as identically zero. In other words,
if the upper limit is below the lower limit, there is no summation.

An inductive proof can be found here (

http://planetmath.org/ProofOfAbelsLemmaByInduction

).
The result can be found in

[

1

]

(Exercise 3.3.5).

If the sequences are indexed from

M

to

N

, we have the following
variant:

Corollary

Let

{

a

i

}

i

=

M

N

and

{

b

i

}

i

=

M

N

be sequences of
real (or complex) numbers with

0

≤

M

≤

N

.
For

n

=

M

,

…

,

N

, let

A

n

be the partial sum

A

n

=

∑

i

=

M

n

a

i

.
Then

∑

i

=

M

N

a

i

⁢

b

i

=

∑

i

=

M

N

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

N

⁢

b

N

.

Proof.

By defining

a

0

=

…

=

a

M

-

1

=

b

0

=

…

=

b

M

-

1

=

0

, we can apply Theorem 1
to the sequences

{

a

i

}

i

=

0

N

and

{

b

i

}

i

=

0

N

.

□

References

1

R.B. Guenther, L.W. Lee,

Partial Differential Equations

of Mathematical Physics and Integral Equations

,
Dover Publications, 1988.

Title

Abel’s lemma

Canonical name

AbelsLemma

Date of creation

2013-03-22 13:19:49

Last modified on

2013-03-22 13:19:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 40A05

Synonym

summation by parts

Synonym

Abel’s partial summation

Synonym

Abel’s

identity

Synonym

Abel’s

transformation

Related topic

PartialSummation

Generated on Fri Feb 9 21:11:18 2018 by

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Abel’s multiplication rule for series

Abel’s multiplication rule for series

Cauchy has originally presented the

multiplication

rule

∑

j

=

1

∞

a

j

⋅

∑

k

=

1

∞

b

k

=

∑

n

=

1

∞

(

a

1

⁢

b

n

+

a

2

⁢

b

n

-

1

+

…

+

a

n

⁢

b

1

)

(1)

for two series.  His

assumption

was that both of the multiplicand series should be

absolutely convergent

.  Mertens (1875) lightened the assumption requiring that both multiplicands should be convergent but at least one of them absolutely convergent (see the parent (

http://planetmath.org/MultiplicationOfSeries

) entry).  N. H. Abel’s most general form of the multiplication rule is the

Theorem

.

The rule (1) for

multiplication of series

with real or complex terms is valid as soon as all three of its series are convergent.

Proof.

We consider the corresponding

power series

∑

j

=

1

∞

a

j

⁢

x

j

,

∑

k

=

1

∞

b

k

⁢

x

k

.

(2)

When

x

=

1

,  they give the series

∑

j

=

1

∞

a

j

,

∑

k

=

1

∞

b

k

which we assume to

converge

.  Thus the power series are absolutely convergent for

|

x

|

<

1

, whence they obey the multiplication rule due to Cauchy:

∑

j

=

1

∞

a

j

⁢

x

j

⋅

∑

k

=

1

∞

b

k

⁢

x

k

=

∑

n

=

1

∞

(

a

1

⁢

b

n

+

a

2

⁢

b

n

-

1

+

…

+

a

n

⁢

b

1

)

⁢

x

n

+

1

.

(3)

On the other hand, the sums of the power series (2) are, as is well known,

continuous functions

on the

interval

[

0

,

1

]

;  the same concerns the right hand side of (3), because for

x

=

1

it becomes the third series which we assume convergent.  When

x

→

1

-

, we infer that

∑

j

=

1

∞

a

j

⁢

x

j

→

∑

j

=

1

∞

a

j

,

∑

k

=

1

∞

b

k

⁢

x

k

→

∑

k

=

1

∞

b

k

and that the limit of the right hand side of (3) is the right hand side of (1).  Since the equation (3) is true for

|

x

|

<

1

,  also the limits of both of (3), as

x

→

1

-

, are equal.  Therefore the equation (1) is in with the assumptions of the theorem.

References

1

E. Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset III. Toinen osa.

Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

Title

Abel’s multiplication rule for series

Canonical

name

AbelsMultiplicationRuleForSeries

Date of creation

2014-11-22 21:19:35

Last modified on

2014-11-22 21:19:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Synonym

Abel’s multiplication rule

Related topic

AbelsLimitTheorem

Related topic

NielsHenrikAbel

Generated on Fri Feb 9 21:11:21 2018 by

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absolute convergence of double series

absolute convergence of double series

Let us consider the

double series

∑

i

,

j

=

1

∞

u

i

⁢

j

(1)

of real or

complex numbers

u

i

⁢

j

.  Denote the

row series

u

k

⁢

1

+

u

k

⁢

2

+

…

by

R

k

, the

column series

u

1

⁢

k

+

u

2

⁢

k

+

…

by

C

k

and the

diagonal series

u

11

+

u

12

+

u

21

+

u

13

+

u

22

+

u

31

+

…

by DS.  Then one has the

Theorem.

All row series, all column series and the diagonal series

converge absolutely

and

∑

k

=

1

∞

R

k

=

∑

k

=

1

∞

C

k

=

DS

,

if one of the following conditions is true:

•

The diagonal series converges absolutely.

•

There exists a positive number

M

such that every finite sum of the numbers

|

u

i

⁢

j

|

is

≦

M

.

•

The row series

R

k

converge absolutely and the series

W

1

+

W

2

+

…

with

∑

j

=

1

∞

|

u

k

⁢

j

|

=

W

k

is convergent.  An analogical condition may be formulated for the column series

C

k

.

Example.

Does the double series

∑

m

=

2

∞

∑

n

=

3

∞

n

-

m

converge

?  If yes, determine its sum.

The column series

∑

m

=

2

∞

(

1

n

)

m

have

positive

terms and are absolutely converging

geometric series

having the sum

(

1

/

n

)

2

1

-

1

/

n

=

1

n

⁢

(

n

-

1

)

=

1

n

-

1

-

1

n

=

W

n

.

The series

W

3

+

W

4

+

…

is convergent, since its

partial sum

is a

telescoping sum

∑

n

=

3

N

W

n

=

∑

n

=

3

N

(

1

n

-

1

-

1

n

)

=

(

1

2

-

1

3

)

+

(

1

3

-

1

4

)

+

(

1

4

-

1

5

)

+

…

+

(

1

N

-

1

-

1

N

)

equalling simply

1

2

-

1

N

and having the limit

1

2

as

N

→

∞

.  Consequently, the given double series converges and its sum is

1

2

.

Title

absolute convergence of double series

Canonical name

AbsoluteConvergenceOfDoubleSeries

Date of creation

2013-03-22 18:46:45

Last modified on

2013-03-22 18:46:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 40A05

Related topic

DoubleSeries

Related topic

DiagonalSumming

Defines

row series

Defines

column series

Defines

diagonal series

Generated on Fri Feb 9 21:11:23 2018 by

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absolute convergence theorem

absolute convergence theorem

Every absolutely convergent series is

convergent

.

Title

absolute convergence theorem

Canonical name

AbsoluteConvergenceTheorem

Date of creation

2013-03-22 13:41:49

Last modified on

2013-03-22 13:41:49

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

4

Author

paolini (1187)

Entry type

Theorem

Classification

msc 40A05

Related topic

RealPartSeriesAndImaginaryPartSeries

Generated on Fri Feb 9 21:11:25 2018 by

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absolutely convergent infinite product converges

absolutely convergent infinite product converges

Theorem.

An

absolutely convergent

(

http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct

) infinite product

∏

ν

=

1

∞

(

1

+

c

ν

)

=

(

1

+

c

1

)

⁢

(

1

+

c

2

)

⁢

(

1

+

c

3

)

⁢

⋯

(1)

of

complex numbers

is convergent.

Proof.

We thus assume the

convergence

of the product (

http://planetmath.org/Product

)

∏

ν

=

1

∞

(

1

+

|

c

ν

|

)

=

(

1

+

|

c

1

|

)

⁢

(

1

+

|

c

2

|

)

⁢

(

1

+

|

c

3

|

)

⁢

⋯

(2)

Let

ε

be an arbitrary positive number.  By the general convergence condition of infinite product, we have

|

(

1

+

|

c

n

+

1

|

)

⁢

(

1

+

|

c

n

+

2

|

)

⁢

⋯

⁢

(

1

+

|

c

n

+

p

|

)

-

1

|

<

ε

∀

p

∈

ℤ

+

when

n

≧

certain

n

ε

.  Then we see that

|

(

1

+

c

n

+

1

)

⁢

(

1

+

c

n

+

2

)

⁢

⋯

⁢

(

1

+

c

n

+

p

)

-

1

|

=

|

1

+

∑

ν

=

n

+

1

n

+

p

c

ν

+

∑

μ

,

ν

c

μ

⁢

c

ν

+

…

+

c

n

+

1

⁢

c

n

+

2

⁢

⋯

⁢

c

n

+

p

-

1

|

≦

1

+

∑

ν

=

n

+

1

n

+

p

|

c

ν

|

+

∑

μ

,

ν

|

c

μ

|

⁢

|

c

ν

|

+

…

+

|

c

n

+

1

|

⁢

|

c

n

+

2

|

⁢

⋯

⁢

|

c

n

+

p

|

-

1

=

|

(

1

+

|

c

n

+

1

|

)

⁢

(

1

+

|

c

n

+

2

|

)

⁢

⋯

⁢

(

1

+

|

c

n

+

p

|

)

-

1

|

<

ε

∀

p

∈

ℤ

+

as soon as

n

≧

n

ε

.  I.e., the infinite product (1)

converges

, by the same convergence condition.

Title

absolutely convergent infinite product converges

Canonical name

AbsolutelyConvergentInfiniteProductConverges

Date of creation

2013-03-22 18:41:15

Last modified on

2013-03-22 18:41:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 30E20

Synonym

convergence of absolutely convergent infinite product

Related topic

AbsoluteConvergenceImpliesConvergenceForAnInfiniteProduct

Related topic

AbsoluteConvergenceOfInfiniteProductAndSeries

Generated on Fri Feb 9 21:11:29 2018 by

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adding and removing parentheses in series

adding and removing parentheses in series

We consider series with real or complex terms.

•

If one groups the terms of a

convergent series

by adding parentheses but not changing the order of the terms, the series remains

convergent

and its sum the same. (See theorem 3 of the

http://planetmath.org/node/6517

parent entry.)

•

A

divergent series

can become convergent if one adds an infinite amount of parentheses; e.g.

1

-

1

+

1

-

1

+

1

-

1

+

-

…

diverges

but

(

1

-

1

)

+

(

1

-

1

)

+

(

1

-

1

)

+

…

converges.

•

A convergent series can become divergent if one removes an infinite amount of parentheses; cf. the preceding example.

•

If a series parentheses, they can be removed if the obtained series converges; in this case also the original series converges and both series have the same sum.

•

If the series

(

a

1

+

…

+

a

r

)

+

(

a

r

+

1

+

…

+

a

2

⁢

r

)

+

(

a

2

⁢

r

+

1

+

…

+

a

3

⁢

r

)

+

…

(1)

converges and

lim

n

→

∞

⁡

a

n

=

0

,

(2)

then also the series

a

1

+

a

2

+

a

3

⁢

…

(3)

converges and has the same sum as (1).

Proof.

Let

S

be the sum of the (1).  Then for each

positive integer

n

, there exists an

integer

k

such that

k

⁢

r

<

n

≦

(

k

+

1

)

⁢

r

.  The

partial sum

of (3) may be written

a

1

+

…

+

a

n

=

(

a

1

+

…

+

a

k

⁢

r

)

⏟

s

+

(

a

k

⁢

r

+

1

+

…

+

a

n

)

⏟

s

′

.

When

n

→

∞

, we have

s

→

S

by the

convergence

of (1) to

S

, and

s

′

→

0

by the condition (2).  Therefore the whole partial sum will tend to

S

, Q.E.D.

Note.

The parenthesis expressions in (1) need not be “equally long” — it suffices that their lengths are under an finite bound.

Title

adding and removing parentheses in series

Canonical name

AddingAndRemovingParenthesesInSeries

Date of creation

2013-03-22 18:54:09

Last modified on

2013-03-22 18:54:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 40A05

Related topic

EmptySum

Generated on Fri Feb 9 21:11:31 2018 by

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almost convergent

almost convergent

A real

sequence

(

x

n

)

is said to be

almost

convergent

to

L

if each

Banach limit

assigns
the same value

L

to the sequence

(

x

n

)

.

Lorentz

[

4

]

proved that

(

x

n

)

is almost convergent to

L

if and only if

lim

p

→

∞

⁡

x

n

+

…

+

x

n

+

p

-

1

p

=

L

uniformly in

n

.

The above limit can be rewritten in detail as

(

∀

ε

>

0

)

(

∃

p

0

)

(

∀

p

>

p

0

)

(

∀

n

)

|

x

n

+

…

+

x

n

+

p

-

1

p

-

L

|

<

ε

.

Almost

convergence

is studied in summability theory. It is an example of a summability method
which cannot be represented as a matrix method.

References

1

G. Bennett and N.J. Kalton:

Consistency theorems

for almost convergence.

Trans. Amer. Math. Soc.

, 198:23–43, 1974.

2

J. Boos:

Classical and modern methods in summability

.
Oxford University Press, New York, 2000.

3

Jeff Connor and K.-G. Grosse-Erdmann:
Sequential definitions of continuity for real functions.

Rocky Mt. J. Math.

, 33(1):93–121, 2003.

4

G. G. Lorentz:
A contribution to the theory of

divergent sequences

.

Acta Math.

, 80:167–190, 1948.

Title

almost convergent

Canonical name

AlmostConvergent

Date of creation

2013-03-22 15:29:51

Last modified on

2013-03-22 15:29:51

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

12

Author

kompik (10588)

Entry type

Definition

Classification

msc 40A05

Classification

msc 40C99

Related topic

Banachlimit

Defines

almost convergent

Generated on Fri Feb 9 21:11:50 2018 by

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alternating series test

alternating series test

The

alternating series test

, or
the

Leibniz’s Theorem

, states the following:

Theorem

[

1

,

2

]

Let

(

a

n

)

n

=

1

∞

be a non-negative, non-increasing

sequence

or

real numbers

such that

lim

n

→

∞

⁡

a

n

=

0

.
Then the

infinite series

∑

n

=

1

∞

(

-

1

)

(

n

+

1

)

⁢

a

n

converges

.

This test provides a necessary and sufficient condition for the

convergence

of an

alternating series

, since if

∑

n

=

1

∞

a

n

converges then

a

n

→

0

.

Example:

The series

∑

k

=

1

∞

1

k

does not converge, but the alternating series

∑

k

=

1

∞

(

-

1

)

k

+

1

⁢

1

k

converges to

ln

⁡

(

2

)

.

References

1

W. Rudin,

Principles of

Mathematical Analysis

, McGraw-Hill Inc., 1976.

2

E. Kreyszig,

Advanced Engineering Mathematics

,
John Wiley & Sons, 1993, 7th ed.

Title

alternating series test

Canonical name

AlternatingSeriesTest

Date of creation

2013-03-22 12:27:09

Last modified on

2013-03-22 12:27:09

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

18

Author

Koro (127)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 40-00

Synonym

Leibniz’s theorem

Synonym

Leibniz test

Related topic

AlternatingSeries

Generated on Fri Feb 9 21:11:51 2018 by

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An Intuitive Insight Into Cross Products

An Intuitive Insight Into Cross Products

An Intuitive Insight Into

Cross Products

Swapnil Sunil Jain 21 December, 2007

Title

An Intuitive Insight Into Cross Products

Canonical

name

AnIntuitiveInsightIntoCrossProducts1

Date of creation

2013-03-11 19:35:32

Last modified on

2013-03-11 19:35:32

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

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