aarjaneiro/planetmath huggingface dataset:
limit of nondecreasing sequence

limit of nondecreasing sequence

Theorem.

A

monotonically nondecreasing

sequence

of

real numbers

with

upper bound

a number

M

converges

to a limit which does not exceed

M

.

Proof.

Let

a

1

≦

a

2

≦

…

≦

a

n

≦

…

≦

M

.  Therefore the set

{

a

1

,

a

2

,

…

}

has a finite supremum

s

≦

M

.  We show that

lim

n

→

∞

⁡

a

n

=

s

.

(1)

Let

ε

an arbitrary positive number.  According to the definition of supremum we have

a

n

≦

s

for all

n

and on the other hand, there exists a member

a

n

⁢

(

ε

)

of the sequence that is

>

s

-

ε

.  Then we have

s

-

ε

<

a

n

⁢

(

ε

)

≦

s

,  and since the sequence is nondecreasing,

0

≦

s

-

a

n

≦

s

-

a

n

⁢

(

ε

)

<

ε

for all

⁢

n

≧

n

⁢

(

ε

)

.

Thus the equation (1) and the whole theorem has been proven.

For the

nonincreasing

sequences there is the corresponding

Theorem.

A monotonically nonincreasing sequence of real numbers with

lower bound

a number

L

converges to a limit which is not less than

L

.

Note.

A good application of the latter theorem is in the proof that Euler’s constant exists.

Title

limit of nondecreasing sequence

Canonical name

LimitOfNondecreasingSequence

Date of creation

2013-03-22 17:40:31

Last modified on

2013-03-22 17:40:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40-00

Synonym

nondecreasing sequence with upper bound

Synonym

limit of increasing sequence

Related topic

MonotonicallyIncreasing

Related topic

MonotoneIncreasing

Related topic

Supremum

Related topic

Infimum

Related topic

ConvergenceOfTheSequence11nn

Generated on Fri Feb 9 21:09:55 2018 by

LaTeXML
limit of sequence as sum of series

limit of sequence as sum of series

If

U

is the limit of a

sequence

u

1

,

u

2

,

u

3

,

…

of real or

complex numbers

, then

U

can be expressed as the series sum

U

=

u

1

+

∑

i

=

1

∞

(

u

i

+

1

-

u

i

)

.

Proof.

Let

s

n

:=

u

1

+

∑

i

=

1

n

-

1

(

u

i

+

1

-

u

i

)

.  We see that

s

n

=

u

1

+

∑

i

=

1

n

-

1

u

i

+

1

-

∑

i

=

1

n

-

1

u

i

=

u

1

+

∑

j

=

2

n

u

j

-

∑

i

=

1

n

-

1

u

i

=

u

n

for all

n

=

1

,

2

,

3

,

…

Thus

u

1

+

∑

i

=

1

∞

(

u

i

+

1

-

u

i

)

=

lim

n

→

∞

⁡

s

n

=

lim

n

→

∞

⁡

u

n

=

U

,

Q.E.D.

Title

limit of sequence as sum of series

Canonical name

LimitOfSequenceAsSumOfSeries

Date of creation

2013-03-22 17:28:21

Last modified on

2013-03-22 17:28:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40-00

Related topic

SumOfSeries

Generated on Fri Feb 9 21:09:57 2018 by

LaTeXML
monotonic

monotonic

A

sequence

or

function

is said to be

monotonic

if it is

•

monotonically increasing

•

monotonically decreasing

•

monotonically nondecreasing

•

monotonically nonincreasing

Intuitively, this means that a monotone sequence can be thought of a “staircase” going either only up, or only down, with the stairs any height and any depth. The same goes for

step functions

. Smooth (

analytic

) monotone functions are like “slopes,” either up or down at potentially varying steepnesess.

Title

monotonic

Canonical name

Monotonic

Date of creation

2013-03-22 12:22:43

Last modified on

2013-03-22 12:22:43

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 40-00

Synonym

monotonically

Synonym

monotone

Generated on Fri Feb 9 21:09:59 2018 by

LaTeXML
monotonically decreasing

monotonically decreasing

A

sequence

(

s

n

)

is

monotonically decreasing

if

s

m

⁢

<

s

n

⁢

∀

m

>

⁢

n

Similarly, a real function

f

⁢

(

x

)

is monotonically decreasing if

f

⁢

(

x

)

⁢

<

f

⁢

(

y

)

⁢

∀

x

>

⁢

y

Compare this to

monotonically nonincreasing

.

Conflict note.

In other context, such as

[

1

]

, this is called

strictly decreasing

. When this is the case, “monotonically nonincreasing” is instead called “monotonically decreasing.”

References

1

“

http://www.nist.gov/dads/HTML/strictlyDecreasing.html

strictly decreasing,” from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.

Title

monotonically decreasing

Canonical name

MonotonicallyDecreasing

Date of creation

2013-03-22 12:22:30

Last modified on

2013-03-22 12:22:30

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 40-00

Synonym

monotone decreasing

Synonym

strictly decreasing

Related topic

MonotonicallyIncreasing

Generated on Fri Feb 9 21:10:00 2018 by

LaTeXML
monotonically increasing

monotonically increasing

A

sequence

(

s

n

)

,

s

n

∈

ℝ

is called

monotonically increasing

if

s

m

>

s

n

⁢

∀

m

>

n

Similarly, a real function

f

⁢

(

x

)

is called monotonically increasing if

f

⁢

(

x

)

>

f

⁢

(

y

)

⁢

∀

x

>

y

Compare this to

monotonically nondecreasing

.

Conflict note.

This condition is also sometimes called

strictly increasing

[

1

]

. In such a context, “monotonically increasing” has the same meaning as monotonically nondecreasing.

References

1

“

http://www.nist.gov/dads/HTML/strictlyIncreasing.html

strictly increasing,” from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.

Title

monotonically increasing

Canonical name

MonotonicallyIncreasing

Date of creation

2013-03-22 12:22:35

Last modified on

2013-03-22 12:22:35

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

10

Author

akrowne (2)

Entry type

Definition

Classification

msc 40-00

Synonym

monotone increasing

Synonym

strictly increasing

Related topic

MonotonicallyDecreasing

Generated on Fri Feb 9 21:10:02 2018 by

LaTeXML
monotonically nondecreasing

monotonically nondecreasing

A

sequence

(

s

n

)

(with real elements) is called

monotonically nondecreasing

if

s

m

≥

s

n

⁢

∀

m

>

n

Similarly, a real function

f

⁢

(

x

)

is called monotonically nondecreasing if

f

⁢

(

x

)

≥

f

⁢

(

y

)

⁢

∀

x

>

y

Compare this to

monotonically increasing

.

Conflict note.

In other contexts, such as

[

1

]

, this is called

monotonically increasing

(despite the fact that the sequence could be “flat.” In such a context, our definition of “monotonically increasing” is called

strictly increasing

.

References

1

“

http://www.nist.gov/dads/HTML/monotoncincr.html

monotonically increasing,” from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.

Title

monotonically nondecreasing

Canonical name

MonotonicallyNondecreasing

Date of creation

2013-03-22 12:22:38

Last modified on

2013-03-22 12:22:38

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 40-00

Synonym

monotone nondecreasing

Related topic

MonotonicallyNonincreasing

Generated on Fri Feb 9 21:10:04 2018 by

LaTeXML
monotonically nonincreasing

monotonically nonincreasing

A

sequence

(

s

n

)

is

monotonically nonincreasing

if

s

m

≤

s

n

⁢

for all

⁢

m

>

n

Similarly, a

real function

f

⁢

(

x

)

is monotonically nonincreasing if

f

⁢

(

x

)

≤

f

⁢

(

y

)

⁢

for all

⁢

x

>

y

Compare this to

monotonically decreasing

.

Conflict note.

In some contexts, such as

[

1

]

, this is called

monotonically decreasing

(in turn, our “monotonically decreasing” is called “

strictly decreasing

”). This is unfortunately counter-intuitive, since a sequence or

function

that is “flat” (such as

f

⁢

(

x

)

=

1

) is somehow “decreasing.” Beware!

1

Examples

•

(

s

n

)

=

1

,

0

,

-

1

,

-

2

,

…

is monotonically nonincreasing. It is also monotonically decreasing.

•

(

s

n

)

=

1

,

1

,

1

,

1

,

…

is nonincreasing but

not

monotonically decreasing.

•

(

s

n

)

=

(

1

n

+

1

)

is nonincreasing (note that

n

is nonnegative).

•

(

s

n

)

=

1

,

1

,

2

,

1

,

1

,

…

is

not

nonincreasing. It also happens to fail to be

monotonically nondecreasing

.

•

(

s

n

)

=

1

,

2

,

3

,

4

,

5

,

…

is

not

nonincreasing, rather it is nondecreasing (and

monotonically increasing

).

References

1

“

http://www.nist.gov/dads/HTML/monotoncdecr.html

monotonically decreasing,” from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.

Title

monotonically nonincreasing

Canonical name

MonotonicallyNonincreasing

Date of creation

2013-03-22 12:22:32

Last modified on

2013-03-22 12:22:32

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 40-00

Synonym

monotone nonincreasing

Synonym

nonincreasing

Related topic

MonotonicallyNondecreasing

Generated on Fri Feb 9 21:10:06 2018 by

LaTeXML
multiplicative cocycle

multiplicative cocycle

Let

f

:

M

→

M

be a measurable

transformation

, and let

μ

be an

invariant

probability measure

. Consider

A

:

M

→

G

⁢

L

⁢

(

d

,

𝐑

)

, a measurable transformation, where GL(d,

R

) is the space of

invertible

square matrices

of size

d

. We define

A

-

1

:

M

→

G

⁢

L

⁢

(

d

,

𝐑

)

by

A

-

1

⁢

(

x

)

=

[

A

⁢

(

x

)

]

-

1

.
Then we define the

sequence

of functions:

ϕ

n

⁢

(

x

)

=

A

⁢

(

f

n

-

1

⁢

(

x

)

)

⁢

⋯

⁢

A

⁢

(

f

⁢

(

x

)

)

⁢

A

⁢

(

x

)

ϕ

-

n

⁢

(

x

)

=

[

ϕ

n

⁢

(

f

-

n

⁢

(

x

)

)

]

-

1

for

n

≥

1

and

x

∈

M

.

It is easy to verify that:

ϕ

m

+

n

⁢

(

x

)

=

ϕ

n

⁢

(

f

m

⁢

(

x

)

)

⁢

ϕ

m

⁢

(

x

)

for

n

,

m

∈

𝐙

and

x

∈

M

.

The sequence

(

ϕ

n

)

n

is called a multiplicative cocycle, or just

cocycle

defined by the transformation

A

.

Title

multiplicative cocycle

Canonical name

MultiplicativeCocycle

Date of creation

2014-03-19 22:13:54

Last modified on

2014-03-19 22:13:54

Owner

Filipe (28191)

Last modified by

Filipe (28191)

Numerical id

4

Author

Filipe (28191)

Entry type

Definition

Synonym

cocycle;

multiplicative

linear cocycle

Related topic

Furstenberg-Kesten theorem

Defines

multiplicative cocycle

Generated on Fri Feb 9 21:10:08 2018 by

LaTeXML
power tower sequence

power tower sequence

For

positive

values of

a

, the

power tower sequence

a

,

a

a

,

a

a

a

,

a

a

a

a

,

…

is

convergent

if and only if

1

e

e

≦

a

≦

e

1

e

,

approximately

0.065989

≦

a

≦

1.444667

.

The limit of the

sequence

is the least real root (

http://planetmath.org/Equation

) of the equation

a

x

=

x

.

The proof is found in [1].

References

1

E. Lindelöf

:

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

Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

Title

power tower sequence

Canonical name

PowerTowerSequence

Date of creation

2013-03-22 16:41:58

Last modified on

2013-03-22 16:41:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 40-00

Related topic

PowerFunction

Related topic

OrderOfOperations

Related topic

NaturalLogBase

Related topic

FunctionXx

Related topic

SuperexponentiationIsNotElementary

Related topic

ErnstLindelof

Defines

power tower sequence

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

LaTeXML
prime theorem of a convergent sequence, a

prime theorem of a convergent sequence, a

Theorem

.

Suppose

(

a

n

)

is a

positive

real

sequence

that

converges

to

L

. Then
the sequence of

arithmetic means

(

b

n

)

=

(

n

-

1

⁢

∑

k

=

1

n

a

k

)

and the sequence of

geometric means

(

c

n

)

=

(

a

1

⁢

⋯

⁢

a

n

n

)

also converge to

L

.

Proof.

We first show that

(

b

n

)

converges to

L

. Let

ε

>

0

. Select a

positive integer

N

0

such that

n

≥

N

0

implies

|

a

n

-

L

|

<

ε

/

2

. Since

(

a

n

)

converges to a finite value, there is a finite

M

such that

|

a

n

-

L

|

<

M

for all

n

. Thus we can select a positive integer

N

≥

N

0

for which

(

N

0

-

1

)

⁢

M

/

N

<

ε

/

2

.

By the

triangle inequality

,

|

b

n

-

L

|

≤

1

n

⁢

∑

k

=

1

n

|

a

k

-

L

|

<

(

N

0

-

1

)

⁢

M

n

+

(

n

-

N

0

+

1

)

⁢

ε

2

⁢

n

<

ε

/

2

+

ε

/

2

.

Hence

(

b

n

)

converges to

L

.

To show that

(

c

n

)

converges to

L

, we first define the sequence

(

d

n

)

by

d

n

=

c

n

n

=

a

1

⁢

⋯

⁢

a

n

. Since

d

n

is a positive real sequence, we have that

lim inf

⁡

d

n

+

1

d

n

≤

lim inf

⁡

d

n

n

≤

lim sup

⁡

d

n

n

≤

lim sup

⁡

d

n

+

1

d

n

,

a proof of which can be found in

[

1

]

. But

d

n

+

1

/

d

n

=

a

n

+

1

, which by

assumption

converges to

L

. Hence

d

n

n

=

c

n

must also converge to

L

.
∎

References

1

Rudin, W.,

Principles of

Mathematical Analysis

, 3rd ed., McGraw-Hill, New York, 1976.

Title

prime

theorem

of a

convergent sequence

, a

Canonical name

PrimeTheoremOfAConvergentSequenceA

Date of creation

2013-03-22 14:49:45

Last modified on

2013-03-22 14:49:45

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

24

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 40-00

Related topic

ArithmeticMean

Related topic

GeometricMean

Generated on Fri Feb 9 21:10:13 2018 by

LaTeXML
proof of limit comparison test

proof of limit comparison test

The main

theorem

we will use is the

comparison test

, which basically states that if

a

n

>

0

,

b

n

>

0

and there is an

N

such that for all

n

>

N

,

a

n

<

b

n

, then if

∑

i

=

1

∞

b

n

converges

so will

∑

i

=

1

∞

a

n

.

Suppose

lim

n

→

∞

⁡

a

n

b

n

=

L

where

L

can be a non negative real number or

+

∞

.

By definition, for

L

finite, this means that for every

ϵ

>

0

there is a

natural number

n

ϵ

such that for all

n

>

n

ϵ

,

∥

a

n

b

n

-

L

∥

<

ϵ

To make matters more concrete choose

ϵ

=

L

2

and assume

L

≠

0

and finite.

0

<

a

n

<

3

⁢

L

2

⁢

b

n

, for all

n

>

n

L

2

.

If

∑

i

=

1

∞

b

n

converges, so will

∑

i

=

1

∞

3

⁢

L

2

⁢

b

n

and thus by the comparison test,

∑

i

=

1

∞

a

n

will also be

convergent

.

For the reverse result, consider

lim

n

→

∞

⁡

b

n

a

n

=

1

L

, since if

L

is finite so will

1

L

, applying the previous result we can say that if

∑

i

=

1

∞

a

n

converges so will

∑

i

=

1

∞

b

n

Consider the case

L

=

0

, clearly

L

=

0

+

since both

a

n

and

b

n

are

positive

, this means that for all

ϵ

>

0

there exists

n

ϵ

such that for all

n

>

n

ϵ

,

0

<

a

n

<

ϵ

⁢

b

n

.

Considering

ϵ

=

1

we get the exact formulation of the comparison test, so if

∑

i

=

1

∞

b

n

converges so will

∑

i

=

1

∞

a

n

.

For the case

L

=

+

∞

just apply the result to

lim

n

→

∞

⁡

b

n

a

n

=

0

to conclude that if

∑

i

=

1

∞

a

n

converges so will

∑

i

=

1

∞

b

n

Title

proof of limit comparison test

Canonical name

ProofOfLimitComparisonTest

Date of creation

2013-03-22 15:35:54

Last modified on

2013-03-22 15:35:54

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

4

Author

cvalente (11260)

Entry type

Proof

Classification

msc 40-00

Generated on Fri Feb 9 21:10:15 2018 by

LaTeXML
Raabe’s criteria

Raabe’s criteria

Theorem

.

The series

a

1

+

a

2

+

a

3

+

⋯

with

positive

is

•

convergent

if, starting from some value of

n

, its fulfil the condition

a

n

+

1

a

n

≦

1

-

μ

n

where

μ

is a and

>

1

;

•

divergent if, starting from some value of

n

, its fulfil the condition

a

n

+

1

a

n

≧

1

-

1

n

-

M

n

2

where

M

is a .

Title

Raabe’s criteria

Canonical name

RaabesCriteria

Date of creation

2013-03-22 15:08:45

Last modified on

2013-03-22 15:08:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40-00

Related topic

ASeriesRelatedToHarmonicSeries

Generated on Fri Feb 9 21:10:31 2018 by

LaTeXML
remainder term

remainder term

Let

S

n

be the

n

th

partial sum

of the series

a

1

+

a

2

+

⋯

with real or complex

a

n

(

n

=

1

,

2

,

…

).

•

If the series is

convergent

with sum

S

, then we call the

R

n

:=

S

-

S

n

the

n

th

remainder term

or simply

remainder

of the series (

n

=

1

,

2

,

…

).  Then

lim

n

→

∞

⁡

R

n

=

0

.

•

If there exists a number

s

such that

lim

n

→

∞

⁡

(

s

-

S

n

)

=

0

,  then the series is convergent and its sum is

s

.

Title

remainder term

Canonical

name

RemainderTerm

Date of creation

2013-03-22 14:51:02

Last modified on

2013-03-22 14:51:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 40-00

Synonym

remainder

Synonym

tail of series

Related topic

SumOfSeries

Generated on Fri Feb 9 21:10:36 2018 by

LaTeXML
remainder term series

remainder term series

For any series

a

1

+

a

2

+

a

3

+

…

(1)

of real or complex terms

a

j

one may interpret its

m

’th

remainder term

R

m

:=

a

m

+

1

+

a

m

+

2

+

…

(2)

as a series.  This

remainder term series

has its own

partial sums

S

m

(

n

)

:=

a

m

+

1

+

a

m

+

2

+

…

+

a

m

+

n

(

n

=

1

,

2

,

…

)

.

(3)

If

m

+

n

=

k

, then the

k

th

partial sum of the original series (1)
is

S

k

=

S

m

+

S

m

(

n

)

.

(4)

For a fixed

m

, the limit

lim

k

→

∞

⁡

S

k

apparently
exists iff the limit

lim

n

→

∞

⁡

S

m

(

n

)

exists.  Thus we can write the

Theorem.

The series (1) is

convergent

if and only if
each remainder term series (2) is convergent.

Cf. the entry ‘‘

finite changes in convergent series

’’.

References

1

Л. Д.
Кудрявцев:

Математический анализ

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

Title

remainder term series

Canonical name

RemainderTermSeries

Date of creation

2014-05-16 21:09:46

Last modified on

2014-05-16 21:09:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 40-00

Generated on Fri Feb 9 21:10:41 2018 by

LaTeXML
series

series

Given a

sequence

of numbers (real or

complex

)

{

a

n

}

we define a sequence of

partial
sums

{

S

N

}

, where

S

N

=

∑

n

=

1

N

a

n

. This sequence is called the

series

with

terms

a

n

. We define the

sum of the series

∑

n

=

1

∞

a

n

to be the limit of these partial sums. More precisely

∑

n

=

1

∞

a

n

=

lim

N

→

∞

⁡

S

n

=

lim

N

→

∞

⁡

∑

n

=

1

N

a

n

.

In a context where this distinction does not matter much (this is usually the case) one identifies a series with its sum, if the latter exists.

Traditionally, as above, series are infinite sums of

real numbers

. However,
the formal constraints on the terms

{

a

n

}

are much less strict. We need
only be able to add the terms and take the limit of partial sums. So in full
generality the terms could be

complex numbers

or

even elements

of certain rings,
fields, and

vector spaces

.

Title

series

Canonical

name

Series

Date of creation

2013-03-22 12:41:35

Last modified on

2013-03-22 12:41:35

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Definition

Classification

msc 40-00

Related topic

AbsoluteConvergence

Related topic

HarmonicNumber

Related topic

CompleteUltrametricField

Related topic

Summation

Related topic

PrimeHarmonicSeries

Generated on Fri Feb 9 21:10:43 2018 by

LaTeXML
sum of series

sum of series

If a series

∑

n

=

1

∞

a

n

of real or

complex
numbers

is

convergent

and the limit of its

partial sums

is

S

,
then

S

is said to be the

sum of the series

.  This
circumstance may be denoted by

∑

n

=

1

∞

a

n

=

S

or equivalently

a

1

+

a

2

+

a

3

+

…

=

S

.

The sum of series has the

distributive property

c

⁢

(

a

1

+

a

2

+

a

3

+

…

)

=

c

⁢

a

1

+

c

⁢

a

2

+

c

⁢

a

3

+

…

with respect to

multiplication

.  Nevertheless, one must not
think that the sum series means an

addition

of infinitely many
numbers — it’s only a question of the limit

lim

n

→

∞

⁡

(

a

1

+

a

2

+

…

+

a

n

)

⏟

partial sum

.

See also the entry “

manipulating convergent series

”!

The sum of the series is equal to the sum of a partial sum and
the corresponding

remainder term

.

Title

sum of series

Canonical name

SumOfSeries

Date of creation

2014-02-15 19:17:15

Last modified on

2014-02-15 19:17:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 40-00

Related topic

SumFunctionOfSeries

Related topic

ManipulatingConvergentSeries

Related topic

RemainderTerm

Related topic

RealPartSeriesAndImaginaryPartSeries

Related topic

LimitOfSequenceAsSumOfSeries

Related topic

PlusSign

Defines

partial sum

Generated on Fri Feb 9 21:10:45 2018 by

LaTeXML
topic entry on analysis

topic entry on analysis

This is an entry-in-progress for a new

topic entry on analysis

.

1.

Topic entry on real numbers

(

http://planetmath.org/TopicEntryOnRealNumbers

)

2.

Topics on calculus

3.

Topic entry on complex analysis

4.

Topic entry on series of complex terms

5.

Fundamental theorems in complex analysis

6.

Topic on several complex variables

7.

Potential theory

8.

A list of common

http://planetmath.org/node/5556

function spaces

9.

Measure theory

10.

Ergodic theory

11.

Convex analysis

12.

Functional analysis

13.

Operator theory

14.

Harmonic analysis

15.

Non-Newtonian

calculus

16.

Bibliography for real analysis

17.

Index of special functions

Title

topic entry on analysis

Canonical name

TopicEntryOnAnalysis

Date of creation

2013-03-22 18:00:13

Last modified on

2013-03-22 18:00:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Topic

Classification

msc 40-00

Classification

msc 30-00

Classification

msc 28-00

Classification

msc 26-00

Synonym

analysis

Synonym

mathematical analysis

Generated on Fri Feb 9 21:10:54 2018 by

LaTeXML
trick to sum all the reciprocal triangular numbers

trick to sum all the reciprocal triangular numbers

The following trick to sum all the

reciprocals

of the

triangular numbers

is funny:

σ

=

1

+

1

3

+

1

6

+

1

10

+

1

15

+

1

21

+

1

28

+

1

36

+

1

45

+

1

55

+

1

66

+

1

78

+

…

=

1

+

(

1

3

+

1

6

)

+

(

1

10

+

1

15

)

+

(

1

21

+

1

28

)

+

(

1

36

+

1

45

)

+

(

1

55

+

1

66

)

+

…

=

1

+

1

2

+

1

2

⋅

1

3

+

1

2

⋅

1

6

+

1

2

⋅

1

10

+

1

2

⋅

1

15

+

…

=

1

+

1

2

⁢

(

1

+

1

3

+

1

6

+

1

10

+

1

15

+

…

)

which implies

σ

=

1

+

σ

/

2

and hence

1

+

1

3

+

1

6

+

1

10

+

1

15

+

1

21

+

1

28

+

1

36

+

1

45

+

1

55

+

1

66

+

1

78

+

…

=

2

Title

trick to sum all the reciprocal triangular numbers

Canonical

name

TrickToSumAllTheReciprocalTriangularNumbers

Date of creation

2013-03-22 18:58:21

Last modified on

2013-03-22 18:58:21

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

6

Author

juanman (12619)

Entry type

Result

Classification

msc 40-00

Classification

msc 11A99

Related topic

TriangularNumbers

Generated on Fri Feb 9 21:10:57 2018 by

LaTeXML