aarjaneiro/planetmath huggingface dataset:
example of summation by parts

example of summation by parts

Proposition

.

The series

∑

n

=

1

∞

sin

⁡

n

⁢

φ

n

and

∑

n

=

1

∞

cos

⁡

n

⁢

φ

n

converge

for every complex value

φ

which is not an

even multiple

of

π

.

Proof.

Let

ε

be an arbitrary positive number. One uses the

sin

⁡

φ

+

sin

⁡

2

⁢

φ

+

…

+

sin

⁡

n

⁢

φ

=

sin

⁡

(

n

+

1

2

)

⁢

φ

-

sin

⁡

φ

2

2

⁢

sin

⁡

φ

2

,

(1)

cos

⁡

φ

+

cos

⁡

2

⁢

φ

+

…

+

cos

⁡

n

⁢

φ

=

-

cos

⁡

(

n

+

1

2

)

⁢

φ

+

cos

⁡

φ

2

2

⁢

sin

⁡

φ

2

,

(2)

proved in the entry “example of

telescoping sum

(

http://planetmath.org/ExampleOfTelescopingSum

)”. These give the

|

sin

⁡

φ

+

sin

⁡

2

⁢

φ

+

…

+

sin

⁡

n

⁢

φ

|

≦

2

2

⁢

|

sin

⁡

φ

2

|

:=

K

φ

,

|

cos

⁡

φ

+

cos

⁡

2

⁢

φ

+

…

+

cos

⁡

n

⁢

φ

|

≦

2

2

⁢

|

sin

⁡

φ

2

|

:=

K

φ

for any

n

=

1

,

2

,

3

,

…

.
We want to apply to the series

∑

n

=

1

∞

cos

⁡

n

⁢

φ

n

the Cauchy general

convergence

criterion (

http://planetmath.org/CauchyCriterionForConvergence

) for series. Let us use here the short notation

cos

N

φ

+

cos

(

N

+

1

)

φ

+

…

+

cos

(

N

+

p

)

φ

:=

S

N

,

N

+

p

(

p

=

0

,

1

,

2

,

…

)

.

Then, utilizing Abel’s

summation by parts

, we obtain

|

∑

n

=

N

N

+

P

cos

⁡

n

⁢

φ

n

|

=

|

∑

p

=

0

P

1

N

+

p

⁢

cos

⁡

(

N

+

p

)

⁢

φ

|

=

|

∑

p

=

0

P

-

1

(

1

N

+

p

-

1

N

+

p

+

1

)

⁢

S

N

,

N

+

p

+

1

N

+

P

⁢

S

N

,

N

+

P

|

≦

≦

∑

p

=

0

P

-

1

(

1

N

+

p

-

1

N

+

p

+

1

)

⁢

|

S

N

,

N

+

P

|

+

1

N

+

P

⁢

|

S

N

,

N

+

P

|

<

<

∑

p

=

0

P

-

1

(

1

N

+

p

-

1

N

+

p

+

1

)

⋅

2

⁢

K

φ

+

1

N

+

P

⋅

2

⁢

K

φ

=

1

N

⋅

2

⁢

K

φ

;

the last form is gotten by telescoping (

http://planetmath.org/TelescopingSum

) the preceding sum and before that by using the

identity

S

N

,

N

+

p

=

[

cos

⁡

φ

+

cos

⁡

2

⁢

φ

+

…

+

cos

⁡

(

N

+

p

)

⁢

φ

]

-

[

cos

⁡

φ

+

cos

⁡

2

⁢

φ

+

…

+

cos

⁡

(

N

-

1

)

⁢

φ

]

.

Thus we see that

|

∑

n

=

N

N

+

P

cos

⁡

n

⁢

φ

n

|

<

2

⁢

K

φ

N

<

ε

for all

natural numbers

P

as soon as

N

>

2

⁢

K

φ

ε

. According to the

Cauchy criterion

, the latter series is

convergent

for the mentioned values of

φ

. The former series is handled similarly.

Title

example of summation by parts

Canonical name

ExampleOfSummationByParts

Date of creation

2013-03-22 17:27:56

Last modified on

2013-03-22 17:27:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 40A05

Related topic

ExampleOfTelescopingSum

Related topic

SineIntegralInInfinity

Related topic

ExampleOfSolvingTheHeatEquation

Generated on Fri Feb 9 21:13:08 2018 by

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example of telescoping sum

example of telescoping sum

Some trigonometric sums, as

∑

k

=

1

n

cos

⁡

k

⁢

α

and

∑

k

=

1

n

sin

⁡

k

⁢

α

, may be telescoped if the terms are first edited by a suitable

goniometric formula

(

http://planetmath.org/GoniometricFormulae

) (‘‘product formula’’). E.g. we may write:

∑

k

=

1

n

cos

⁡

k

⁢

α

=

1

sin

⁡

α

2

⁢

∑

k

=

1

n

cos

⁡

k

⁢

α

⁢

sin

⁡

α

2

The product formula

cos

⁡

x

⁢

sin

⁡

y

=

1

2

⁢

[

sin

⁡

(

x

+

y

)

-

sin

⁡

(

x

-

y

)

]

alters this to

∑

k

=

1

n

cos

⁡

k

⁢

α

=

1

2

⁢

sin

⁡

α

2

⁢

∑

k

=

1

n

(

sin

⁡

(

2

⁢

k

+

1

)

⁢

α

2

-

sin

⁡

(

2

⁢

k

-

1

)

⁢

α

2

)

,

or

∑

k

=

1

n

cos

k

α

=

1

2

⁢

sin

⁡

α

2

(

sin

3

⁢

α

2

-

sin

α

2

+

sin

5

⁢

α

2

-

sin

3

⁢

α

2

+

-

…

+

sin

(

2

⁢

n

+

1

)

⁢

α

2

-

sin

(

2

⁢

n

-

1

)

⁢

α

2

)

.

After cancelling the opposite numbers we obtain the formula

∑

k

=

1

n

cos

⁡

k

⁢

α

=

sin

⁡

(

2

⁢

n

+

1

)

⁢

α

2

-

sin

⁡

α

2

2

⁢

sin

⁡

α

2

.

(1)

The corresponding formula

∑

k

=

1

n

sin

⁡

k

⁢

α

=

-

cos

⁡

(

2

⁢

n

+

1

)

⁢

α

2

+

cos

⁡

α

2

2

⁢

sin

⁡

α

2

.

(2)

is derived analogously.

Note.

The formulae (1) and (2) are gotten also by adding the left side of the former and

i

times the left side of the latter and then applying

de Moivre identity

.

References

1

Л. Д. Кудрявцев:

Математический анализ. II том

.  Издательство
‘‘Высшая школа’’. Москва (1970).

Title

example of telescoping sum

Canonical name

ExampleOfTelescopingSum

Date of creation

2013-03-22 17:27:21

Last modified on

2013-03-22 17:27:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Example

Classification

msc 40A05

Related topic

GoniometricFormulae

Related topic

ExampleOfSummationByParts

Related topic

DirchletKernel

Generated on Fri Feb 9 21:13:14 2018 by

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example using Stolz-Cesaro theorem

example using Stolz-Cesaro theorem

Example:

We try to determine the value of

lim

n

→

∞

⁡

1

k

+

2

k

+

…

+

n

k

n

k

+

1

,

k

∈

ℕ

.

We consider the

sequences

α

n

≥

1

=

1

k

+

2

k

+

…

+

n

k

and

β

n

≥

1

=

n

k

and using the

Stolz-Cesaro theorem

we have that

lim

n

→

∞

⁡

1

k

+

2

k

+

…

+

n

k

n

k

+

1

=

(1)

lim

n

→

∞

⁡

(

1

k

+

2

k

+

…

+

(

n

+

1

)

k

)

-

(

1

k

+

2

k

+

…

+

n

k

)

(

n

+

1

)

k

+

1

-

n

k

+

1

=

(2)

lim

n

→

∞

⁡

(

n

+

1

)

k

(

n

+

1

)

k

+

1

-

n

k

+

1

.

(3)

Now we try to get the expression in the

indeterminate
form

0

0

as n approaches

∞

, dividing numerator and denominator of
(3) by

(

n

+

1

)

k

.

lim

n

→

∞

⁡

1

(

n

+

1

)

-

n

k

+

1

⁢

(

n

+

1

)

-

k

=

(4)

lim

n

→

∞

⁡

1

n

⁢

(

1

+

n

-

1

-

(

1

+

n

-

1

)

-

k

)

=

(5)

lim

n

→

∞

⁡

n

-

1

1

+

n

-

1

-

(

1

+

n

-

1

)

-

k

.

(6)

By applying L’Hôpital’s rule once we get

lim

n

→

∞

⁡

n

-

1

1

+

n

-

1

-

(

1

+

n

-

1

)

-

k

=

(7)

lim

n

→

∞

⁡

-

n

-

2

-

n

-

2

-

k

⁢

(

1

+

n

-

1

)

-

k

-

1

⁢

n

-

2

=

(8)

lim

n

→

∞

⁡

1

1

+

k

⁢

(

1

+

n

-

1

)

-

k

-

1

=

(9)

1

1

+

k

.

(10)

Title

example using Stolz-Cesaro theorem

Canonical name

ExampleUsingStolzCesaroTheorem

Date of creation

2013-03-22 15:31:02

Last modified on

2013-03-22 15:31:02

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

4

Author

georgiosl (7242)

Entry type

Example

Classification

msc 40A05

Related topic

StolzCesaroTheorem

Related topic

LHpitalsRule

Generated on Fri Feb 9 21:13:18 2018 by

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finite changes in convergent series

finite changes in convergent series

The following

theorem

means that at the beginning of a

convergent series

, one can remove or attach a finite amount of terms without influencing on the

convergence

of the series – the convergence is determined alone by the infinitely long “tail” of the series.  Consequently, one can also freely change the of a finite amount of terms.

Theorem.

Let

k

be a

natural number

.  A series

∑

n

=

1

∞

a

n

converges

iff
the series

∑

n

=

k

+

1

∞

a

n

converges.  Then the sums of both series are with

∑

n

=

k

+

1

∞

a

n

=

∑

n

=

1

∞

a

n

-

∑

n

=

1

k

a

n

.

(1)

Proof.

Denote the

k

th

partial sum

of

∑

n

=

1

∞

a

n

by

S

k

and the

n

th partial sum of

∑

n

=

k

+

1

∞

a

n

by

S

n

′

.  Then we have

S

n

′

=

∑

n

=

k

+

1

k

+

n

a

n

=

S

k

+

n

-

S

k

.

(2)

1

∘

.  If

∑

n

=

1

∞

a

n

converges, i.e.

lim

n

→

∞

⁡

S

n

:=

S

exists as a finite number, then (2) implies

lim

n

→

∞

⁡

S

n

′

=

lim

n

→

∞

⁡

S

k

+

n

-

lim

n

→

∞

⁡

S

k

=

S

-

S

k

.

Thus

∑

n

=

k

+

1

∞

a

n

converges and (1) is true.

2

∘

.  If we suppose

∑

n

=

k

+

1

∞

a

n

to be

convergent

, i.e.

lim

n

→

∞

⁡

S

n

′

:=

S

′

exists as finite, then (2) implies that

lim

n

→

∞

⁡

S

n

=

lim

n

→

∞

⁡

S

k

+

n

=

lim

n

→

∞

⁡

(

S

k

+

S

n

′

)

=

S

k

+

S

′

.

This means that

∑

n

=

1

∞

a

n

is convergent and

S

=

S

k

+

S

′

,  which is (1), is in .

Title

finite changes in convergent series

Canonical name

FiniteChangesInConvergentSeries

Date of creation

2013-03-22 19:03:10

Last modified on

2013-03-22 19:03:10

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Related topic

SumOfSeriesDependsOnOrder

Related topic

RiemannSeriesTheorem

Related topic

RatioTestOfDAlembert

Generated on Fri Feb 9 21:13:28 2018 by

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Forum Policy

Forum Policy

Forum Policy

1

Synopsis

This brief document summarizes rules/regulations, conventions, and tips for using the

PlanetMath

forums.

2

Rules

The rules for the PlanetMath forums are:

1.

The forums are, for the most part, to be “G-rated.” So, no profanity, please. The only possible exception to the G-rated rule is the “Humor” forum, but please keep it to “PG-13” level here.

2.

No flaming! How do you know if you are flaming? Basically, if you are arguing, not debating. Don’t get emotional in the forums!

3.

No spamming/commercial advertising! If you want to advertise on this site, buy Google ads. We keep the commercial and noncommercial stuff cleanly separated here.

As you can see, the rules are few, and are pretty intuitive. However it is important to understand what makes these “rules” and not “suggestions”:

if you break them, you are subject to enforcement

on this site. That means possibly getting locked out, having your account deleted and/or banned, and possibly losing control of any objects you own on the system. Enforcement is at the discretion of the administration!

3

Guidelines

This section is for “soft” forum policy: guidelines, tips, and conventions. It is highly recommended that you follow them, but there is no administrative enforcement.

•

When asking for help on mathematics,

do not ask for the answer

. Nobody wants to do your work for you! Instead, state the problem and what you are confused about, and someone will almost certainly help you through it.

•

No “shilling.” Put another way, shamelessly self-(or otherwise) promoting advertising is frowned upon. However, it is ok to forward the bulletins of noncommercial math societies and the like.

•

Be polite, even when other people aren’t! Be the more mature person! Messages that don’t violate the rules will never go away; don’t say anything you’ll look back on and be ashamed of (of course, its ok and natural to be

wrong

, but don’t be

mean

).

4

Updates

Your help in developing this policy is welcome. Of course the administration has the last word on the rules, but feel free to add to the guidelines. Discussion of proposed changes to this document in the accompanying forum is recommended.

Title

Forum Policy

Canonical

name

ForumPolicy1

Date of creation

2013-03-11 19:22:35

Last modified on

2013-03-11 19:22:35

Owner

akrowne (2)

Last modified by

(0)

Numerical id

1

Author

akrowne (0)

Entry type

Definition

Generated on Fri Feb 9 21:13:30 2018 by

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geometric series

geometric series

A

geometric series

is a series of the form

∑

i

=

1

n

a

⁢

r

i

-

1

(with

a

and

r

real or

complex numbers

). The

partial sums

of a geometric series are given by

s

n

=

∑

i

=

1

n

a

⁢

r

i

-

1

=

a

⁢

(

1

-

r

n

)

1

-

r

.

(1)

An

infinite geometric series

is a geometric series, as above, with

n

→

∞

. It is denoted by

∑

i

=

1

∞

a

⁢

r

i

-

1

If

|

r

|

≥

1

, the infinite geometric series

diverges

. Otherwise it converges to

∑

i

=

1

∞

a

⁢

r

i

-

1

=

a

1

-

r

(2)

Taking the limit of

s

n

as

n

→

∞

, we see that

s

n

diverges if

|

r

|

≥

1

. However, if

|

r

|

<

1

,

s

n

approaches (2).

One way to prove (1) is to take

s

n

=

a

+

a

⁢

r

+

a

⁢

r

2

+

⋯

+

a

⁢

r

n

-

1

and multiply by

r

, to get

r

⁢

s

n

=

a

⁢

r

+

a

⁢

r

2

+

a

⁢

r

3

+

⋯

+

a

⁢

r

n

-

1

+

a

⁢

r

n

subtracting the two removes most of the terms:

s

n

-

r

⁢

s

n

=

a

-

a

⁢

r

n

factoring and dividing gives us

s

n

=

a

⁢

(

1

-

r

n

)

1

-

r

□

Title

geometric series

Canonical name

GeometricSeries

Date of creation

2013-03-22 12:05:37

Last modified on

2013-03-22 12:05:37

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 40A05

Related topic

GeometricSequence

Related topic

ExampleOfAnalyticContinuation

Related topic

ApplicationOfCauchyCriterionForConvergence

Defines

infinite geometric series

Generated on Fri Feb 9 21:13:32 2018 by

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golden ratio

golden ratio

The “

Golden Ratio

”, or

ϕ

, has the value

1.61803398874989484820

⁢

…

This number gets its rather illustrious name from the fact that the Greeks thought that a

rectangle

with ratio of side lengths equal to

ϕ

was the most pleasing to the eye, and much of classical Greek architecture is based on this

premise

. In , an aesthetically pleasing aspect of a rectangle with this ratio, from a mathematical viewpoint, is that if we embed and remove a

w

×

w

square in the below diagram, the remaining rectangle also has a width-to-length ratio of

ϕ

.

Above: The golden rectangle;

l

/

w

=

ϕ

.

ϕ

has plenty of interesting mathematical , however. Its value is exactly

1

+

5

2

The value

1

-

5

2

is often called

ϕ

′

.

ϕ

and

ϕ

′

are the two roots of the

recurrence relation

given by the

Fibonacci sequence

. The following hold for

ϕ

and

ϕ

′

:

•

1

ϕ

=

-

ϕ

′

•

1

-

ϕ

=

ϕ

′

•

1

ϕ

′

=

-

ϕ

•

1

-

ϕ

′

=

ϕ

and so on. These give us

ϕ

-

1

+

ϕ

0

=

ϕ

1

which implies

ϕ

n

-

1

+

ϕ

n

=

ϕ

n

+

1

Title

golden ratio

Canonical

name

GoldenRatio

Date of creation

2013-03-22 11:56:02

Last modified on

2013-03-22 11:56:02

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

17

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 40A05

Classification

msc 11B39

Synonym

golden number

Related topic

ProportionEquation

Related topic

ConstructionOfCentralProportion

Related topic

DerivationOfPlasticNumber

Generated on Fri Feb 9 21:13:35 2018 by

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Halley’s formula

Halley’s formula

The following formula is due to the English scientist and mathematician Edmond Halley (1656 à 1742):

ln

⁡

x

=

lim

n

→

∞

⁡

(

x

n

-

1

)

⁢

n

(1)

Proof.

We change the

n

th root to power of

e

and use the

power series

expansion of

exponential function

:

(

x

n

-

1

)

⁢

n

=

(

e

ln

⁡

x

n

-

1

)

⁢

n

=

(

∑

m

=

0

∞

(

ln

x

/

n

)

m

)

m

!

-

1

)

⁢

n

=

∑

m

=

1

∞

(

ln

⁡

x

/

n

)

m

⁢

n

m

!

=

ln

⁡

x

+

1

n

⁢

∑

m

=

2

∞

(

ln

⁡

x

)

m

m

!

⁢

n

m

-

2

The last converging series has a finite sum, and as

n

→

∞

,  the asserted formula follows.

Note.

The formula (1) was known also by Leonhard Euler, who used it for defining the

natural logarithm

.  Using (1), one can easily prove the well-known laws of

logarithm

, e.g.

ln

⁡

x

⁢

y

=

lim

n

→

∞

⁡

(

x

n

⁢

y

n

-

1

)

⁢

n

=

lim

n

→

∞

⁡

(

x

n

⁢

y

n

-

y

n

+

y

n

-

1

)

⁢

n

=

lim

n

→

∞

⁡

y

1

n

⁢

(

x

n

-

1

)

⁢

n

+

lim

n

→

∞

⁡

(

y

n

-

1

)

⁢

n

=

y

0

⁢

ln

⁡

x

+

ln

⁡

y

=

ln

⁡

x

+

ln

⁡

y

.

References

1

Paul Loya

:

Amazing and Aesthetic Aspect of

Analysis

: On the incredible infinite

. A course in undergraduate analysis, fall 2006. Available

http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdf

here.

Title

Halley’s formula

Canonical name

HalleysFormula

Date of creation

2013-03-22 19:34:39

Last modified on

2013-03-22 19:34:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Result

Classification

msc 40A05

Related topic

ListOfCommonLimits

Generated on Fri Feb 9 21:13:58 2018 by

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Hamiltonian-type elliptic system

Hamiltonian-type elliptic system

Hamiltonian-type elliptic system

A hamiltonian-type elliptic system is of the form

(

H

⁢

S

)

⁢

{

-

Δ

⁢

u

=

G

v

⁢

(

x

;

u

,

v

)

,

x

∈

Ω

-

Δ

⁢

v

=

G

u

⁢

(

x

;

u

,

v

)

,

x

∈

Ω

u

=

v

=

0

⁢

or

⁢

∂

⁡

u

∂

⁡

n

=

∂

⁡

v

∂

⁡

n

=

0

x

∈

∂

⁡

Ω

where

Ω

⊂

ℝ

N

(

N

≥

1

)

is an open bounded domain,

G

⁢

(

x

;

u

,

v

)

∈

𝒞

1

⁢

(

Ω

¯

×

ℝ

2

;

ℝ

)

in the

variables

(

u

,

v

)

∈

ℝ

2

with

∇

⁡

G

=

(

G

u

,

G

v

)

.

Title

Hamiltonian-type elliptic system

Canonical

name

HamiltoniantypeEllipticSystem

Date of creation

2013-03-11 19:28:32

Last modified on

2013-03-11 19:28:32

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

Generated on Fri Feb 9 21:14:00 2018 by

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harmonic series

harmonic series

The

harmonic series

is

h

=

∑

n

=

1

∞

1

n

The harmonic series is known to

diverge

. This can be proven via the

integral test

; compare

h

with

∫

1

∞

1

x

⁢

𝑑

x

.

The harmonic series is a special case of the

p

-series

,

h

p

, which has the form

h

p

=

∑

n

=

1

∞

1

n

p

where

p

is some positive real number. The series is known to converge (leading to the p-series test for series

convergence

) iff

p

>

1

. In using the

comparison test

, one can often compare a given series with

positive

terms to some

h

p

.

Remark 1.

One could call

h

p

with

p

>

1

an

overharmonic series

and

h

p

with

p

<

1

an

underharmonic series

; the corresponding names are known at least in Finland.

Remark 2.

A

p

-series is sometimes called

a

harmonic series, so that

the

harmonic series is a harmonic series with

p

=

1

.

For complex-valued

p

,

h

p

=

ζ

⁢

(

p

)

, the

Riemann zeta function

.

A famous

p

-series is

h

2

(or

ζ

⁢

(

2

)

), which converges to

π

2

6

. In general no

p

-series of odd

p

has been solved analytically.

A

p

-series which is not summed to

∞

, but instead is of the form

h

p

⁢

(

k

)

=

∑

n

=

1

k

1

n

p

is called a

p

-series (or a harmonic series) of order

k

of

p

.

Title

harmonic series

Canonical name

HarmonicSeries

Date of creation

2013-03-22 13:02:46

Last modified on

2013-03-22 13:02:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 40A05

Related topic

HarmonicNumber

Related topic

PrimeHarmonicSeries

Related topic

SumOfPowers

Defines

p-series

Defines

harmonic series of order

Generated on Fri Feb 9 21:14:02 2018 by

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infinite dimensional hamiltonian system

infinite dimensional hamiltonian system

infinite dimensional hamiltonian system

An infinite dimensional Hamiltonian system takes the form

(

I

⁢

H

⁢

S

)

⁢

{

∂

t

⁡

u

-

Δ

x

⁢

u

=

H

v

⁢

(

t

,

x

,

u

,

v

)

-

∂

t

⁡

v

-

Δ

x

⁢

v

=

H

u

⁢

(

t

,

x

,

u

,

v

)

∀

(

t

,

x

)

∈

ℝ

×

Ω

where

Ω

⊂

ℝ

N

is a smoothly bounded domain,

H

∈

𝒞

1

⁢

(

ℝ

×

Ω

¯

×

ℝ

2

,

ℝ

)

.

Title

infinite dimensional hamiltonian system

Canonical

name

InfiniteDimensionalHamiltonianSystem1

Date of creation

2013-03-11 19:28:50

Last modified on

2013-03-11 19:28:50

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

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

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integral test

integral test

Consider a

sequence

(

a

n

)

=

{

a

0

,

a

1

,

a

2

,

a

3

,

…

}

and given

M

∈

ℝ

consider any

monotonically nonincreasing

function

f

:

[

M

,

+

∞

)

→

ℝ

which extends the sequence, i.e.

f

⁢

(

n

)

=

a

n

∀

n

≥

M

An example is

a

n

=

2

⁢

n

→

f

⁢

(

x

)

=

2

⁢

x

(the former being the sequence

{

0

,

2

,

4

,

6

,

8

,

…

}

and the later the doubling function for any

real number

.

We are interested on finding out when the summation

∑

n

=

0

∞

a

n

converges

.

The

integral test

states the following.

The series

∑

n

=

0

∞

a

n

converges if and only if the

integral

∫

M

∞

f

⁢

(

x

)

⁢

𝑑

x

is finite.

Title

integral test

Canonical name

IntegralTest

Date of creation

2013-03-22 12:27:12

Last modified on

2013-03-22 12:27:12

Owner

drini (3)

Last modified by

drini (3)

Numerical id

20

Author

drini (3)

Entry type

Theorem

Classification

msc 40A05

Related topic

Function

Related topic

Sequence

Related topic

Limit

Generated on Fri Feb 9 21:14:13 2018 by

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Kempner series

Kempner series

The

harmonic series

∑

n

=

1

∞

1

n

=

1

1

+

1

2

+

1

3

+

1

4

+

…

(1)

is

divergent

.  The situation is different when one omits from this series all terms whose

denominators

contain in the decimal system (

http://planetmath.org/PositionalNumberSystems

) some digits 9.  Kempner proved 1914 very simply that such a “

depleted harmonic series

” is

convergent

and that its sum is less than 90.  This series is called

Kempner series

.  A better value of the sum with 15 decimals is 22.920676619264150.

The digit 9 here has no special status; one has for other digits

0

,

1

,

2

,

…

,

8

and for digit strings, as “716”.  E.g., we show the

convergence

of the partial series

K

0

⁢

{

=

1

1

+

1

2

+

1

3

+

1

4

+

1

5

+

1

6

+

1

7

+

1

8

+

1

9

+

1

11

+

1

12

+

…

+

1

19

+

1

21

+

…

⁢

…

+

1

99

+

1

111

+

1

112

+

…

+

1

119

+

1

121

+

…

⁢

…

+

1

999

+

.

.

.

.

.

.

of (1) where the denominators contain no 0’s.  Every digit in the denominators has nine possibilities.  For this series we thus get the estimation

K

0

<

9

⋅

1

1

+

9

⋅

9

⋅

1

10

+

9

⋅

9

⋅

9

⋅

1

100

+

…

=

9

+

9

⋅

9

10

+

9

⋅

(

9

10

)

2

+

…

=

9

1

-

9

10

=

90

(a sum of convergent

geometric series

).

For determining more accurately the sums of depleted harmonic series (cf.

http://en.wikipedia.org/wiki/Depleted_uranium

depleted uranium), see the article

http://arxiv.org/ftp/arxiv/papers/0806/0806.4410.pdf

Summing

the curious series of Kempner and Irwin of Baillie.

Title

Kempner series

Canonical name

KempnerSeries

Date of creation

2013-03-22 19:12:59

Last modified on

2013-03-22 19:12:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Result

Classification

msc 40A05

Synonym

depleted harmonic series

Generated on Fri Feb 9 21:14:22 2018 by

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Kronecker’s lemma

Kronecker’s lemma

Kronecker’s lemma gives a condition for

convergence

of

partial sums

of real numbers, and for example can be used in the proof of Kolmogorov’s

strong law of large numbers

.

Lemma

(Kronecker).

Let

x

1

,

x

2

,

…

and

0

<

b

1

<

b

2

<

⋯

be

sequences

of real numbers such that

b

n

increases to

infinity

as

n

→

∞

. Suppose that the sum

∑

n

=

1

∞

x

n

/

b

n

converges

to a finite limit. Then,

b

n

-

1

⁢

∑

k

=

1

n

x

k

→

0

as

n

→

∞

.

Proof.

Set

u

n

=

∑

k

=

1

n

x

k

/

b

k

, so that the limit

u

∞

=

lim

n

→

∞

⁡

u

n

exists.
Also set

a

n

=

∑

k

=

1

n

-

1

(

b

k

+

1

-

b

k

)

⁢

u

k

so that

a

n

+

1

-

a

n

b

n

+

1

-

b

n

=

u

n

→

u

∞

as

n

→

∞

. Then, the

Stolz-Cesaro theorem

says that

a

n

/

b

n

also converges to

u

∞

, so

b

n

-

1

⁢

∑

k

=

1

n

x

k

=

b

n

-

1

⁢

∑

k

=

1

n

b

k

⁢

(

u

k

-

u

k

-

1

)

=

u

n

-

b

n

-

1

⁢

a

n

→

0

.

∎

Title

Kronecker’s lemma

Canonical name

KroneckersLemma

Date of creation

2013-03-22 18:33:54

Last modified on

2013-03-22 18:33:54

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 40-00

Related topic

StolzCesaroTheorem

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