aarjaneiro/planetmath huggingface dataset:
Lane-Emden System

Lane-Emden System

Lane-Emden System

Here is an example of Hamiltonian elliptic systems,
also called the Lane-Emden system,

(

L

⁢

E

)

⁢

{

-

Δ

⁢

u

=

v

p

x

∈

Ω

,

-

Δ

⁢

v

=

u

q

x

∈

Ω

,

u

=

v

=

0

x

∈

∂

⁡

Ω

,

where

p

,

q

>

0

,

Ω

⊂

ℝ

N

(

N

≥

1

)

is an open bounded domain. (LE) is called sublinear (superlinear)
if

p

q

<

1

(

p

q

>

1

)

.
The associated energy

functional

to (LE) is

J

⁢

(

u

,

v

)

=

∫

Ω

∇

⁡

u

⁢

∇

⁡

v

⁢

d

⁢

x

-

∫

Ω

(

1

q

+

1

⁢

u

q

+

1

+

1

p

+

1

⁢

v

p

+

1

)

⁢

𝑑

x

.

Title

Lane-Emden System

Canonical

name

LaneEmdenSystem

Date of creation

2013-03-11 19:34:47

Last modified on

2013-03-11 19:34:47

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

Generated on Fri Feb 9 21:14:26 2018 by

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Leibniz’ estimate for alternating series

Leibniz’ estimate for alternating series

Theorem (Leibniz 1682).

If

p

1

>

p

2

>

p

3

>

⋯

and

lim

m

→

∞

⁡

p

m

=

0

,  then the

alternating series

p

1

-

p

2

+

p

3

-

p

4

+

-

…

(1)

converges

.  Its

remainder term

has the same sign (

http://planetmath.org/SignumFunction

) as the first omitted

±

p

m

+

1

and the

absolute value

less than

p

m

+

1

.

Proof.

The

convergence

of (1) is proved
here (

http://planetmath.org/ProofOfAlternatingSeriesTest

).  Now denote the sum of the series by

S

and the

partial sums

of it by

S

1

,

S

2

,

S

3

,

…

.  Suppose that (1) is truncated after a

negative

-

p

2

⁢

n

.  Then the remainder term

R

2

⁢

n

=

S

-

S

2

⁢

n

may be written in the form

R

2

⁢

n

=

(

p

2

⁢

n

+

1

-

p

2

⁢

n

+

2

)

+

(

p

2

⁢

n

+

3

-

p

2

⁢

n

+

4

)

+

…

or

R

2

⁢

n

=

p

2

⁢

n

+

1

-

(

p

2

⁢

n

+

2

-

p

2

⁢

n

+

3

)

-

(

p

2

⁢

n

+

4

-

p

2

⁢

n

+

5

)

-

…

The former shows that

R

2

⁢

n

is

positive

as the first omitted

p

2

⁢

n

+

1

and the latter that

|

R

2

⁢

n

|

<

p

2

⁢

n

+

1

.  Similarly one can see the assertions true when the series (1) is truncated after a positive

p

2

⁢

n

-

1

.

A pictorial proof.

As seen in this diagram, whenever

m

′

>

m

,  we have

|

S

m

′

-

S

m

|

≤

p

m

+

1

→

0

.  Thus the partial sums form a

Cauchy sequence

, and hence converge.  The limit lies in the of the spiral, strictly in

S

m

and

S

m

+

1

for any

m

.  So the

remainder

after the

m

th must have the same direction as

±

p

m

+

1

=

S

m

+

1

-

S

m

and lesser magnitude.

Example 1.

The alternating series

1

2

-

1

-

1

2

+

1

+

1

3

-

1

-

1

3

+

1

+

1

4

-

1

-

1

4

+

1

+

-

…

does not fulfil the requirements of the theorem and is

divergent

.

Example 2.

The alternating series

1

ln

⁡

2

-

1

ln

⁡

3

+

1

ln

⁡

4

-

1

ln

⁡

5

+

-

…

satisfies all conditions of the theorem and is

convergent

.

Title

Leibniz’ estimate for alternating series

Canonical name

LeibnizEstimateForAlternatingSeries

Date of creation

2014-07-22 15:34:38

Last modified on

2014-07-22 15:34:38

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

35

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 40-00

Synonym

Leibniz’ estimate for remainder term

Related topic

EIsIrrational2

Related topic

ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions

Generated on Fri Feb 9 21:14:54 2018 by

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Loop theorem

Loop theorem

Loop theorem

THE LOOP THEOREM

MSC 57M35

In the

topology

of 3-manifolds,

the loop theorem

is

generalization

of an ansatz discovered by Max Dehn (The Dehn’s Lemma),
who saw that

if a

continuous map

from a 2-disk to a 3-manifold whose

restriction

to the

boundary

’s disk has no singularities,
then it exists another

embedding

whose restriction to the boundary’s disk is equal to the boundary’s restriction original map

.

The following statement called the Loop Theorem is a version from J. Stalling, but written in W. Jaco’s book.

Let

M

be a three-manifold and let

S

be a connected surface in

∂

⁡

M

. Let

N

⊂

π

(

M

)

be a

normal subgroup

.
Let

f

:

D

2

→

M

be a

continuous map

such that

f

⁢

(

∂

⁡

D

2

)

⊂

S

and

[

f

|

∂

D

2

]

∉

N

.

Then there exists an

embedding

g

:

D

2

→

M

such that

g

⁢

(

∂

⁡

D

2

)

⊂

S

and

[

g

|

∂

D

2

]

∉

N

,

The proof is a clever construction due to C. Papakyriakopoulos about a

sequence

(a tower) of covering spaces.
Maybe the best detailed redaction is due to A. Hatcher.
But in general, accordingly to Jaco’s opinion,

”… for anyone unfamiliar with the techniques of 3-manifold-topology and are here to gain a working knowledge for the study of problems in this area…, there is no better place to start.”

==

References

==

W. Jaco,

Lectures on 3-manifolds topology

, A.M.S. regional conference series in Math 43.

J. Hempel,

3-manifolds

, Princeton University Press 1976.

A. Hatcher,

Notes on 3-manifolds

,

Title

Loop theorem

Canonical

name

LoopTheorem12

Date of creation

2013-03-11 19:24:08

Last modified on

2013-03-11 19:24:08

Owner

juanman (12619)

Last modified by

(0)

Numerical id

1

Author

juanman (0)

Entry type

Definition

Generated on Fri Feb 9 21:14:59 2018 by

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manipulating convergent series

manipulating convergent series

The of the series in the following theorems are supposed to be either real or

complex numbers

.

Theorem 1

.

If the series

a

1

+

a

2

+

⋯

and

b

1

+

b

2

+

⋯

converge

and have the sums

a

and

b

, respectively, then also the series

(

a

1

+

b

1

)

+

(

a

2

+

b

2

)

+

⋯

(1)

converges and has the sum

a

+

b

.

Proof.

The

n

th

partial sum

of (1) has the limit

lim

n

→

∞

⁡

∑

j

=

1

n

(

a

j

+

b

j

)

=

lim

n

→

∞

⁡

∑

j

=

1

n

a

j

+

lim

n

→

∞

⁡

∑

j

=

1

n

b

j

=

a

+

b

.

Theorem 2

.

If the series

a

1

+

a

2

+

⋯

converges having the sum

a

and if

c

is any , then also the series

c

⁢

a

1

+

c

⁢

a

2

+

⋯

(2)

converges and has the sum

c

⁢

a

.

Proof.

The

n

th

partial sum of (2) has the limit

lim

n

→

∞

⁡

∑

j

=

1

n

c

⁢

a

j

=

c

⁢

lim

n

→

∞

⁡

∑

j

=

1

n

a

j

=

c

⁢

a

.

Theorem 3

.

If the of any converging series

a

1

+

a

2

+

a

3

+

⋯

(3)

are grouped arbitrarily without changing their , then the resulting series

(

a

1

+

⋯

+

a

m

1

)

+

(

a

m

1

+

1

+

⋯

+

a

m

2

)

+

(

a

m

2

+

1

+

⋯

+

a

m

3

)

+

⋯

(4)

also converges and its sum equals to the sum of (3).

Proof.

Since all the partial sums of (4) are simultaneously partial sums of (3), they have as limit the sum of the series (3).

Title

manipulating convergent series

Canonical name

ManipulatingConvergentSeries

Date of creation

2013-03-22 14:50:49

Last modified on

2013-03-22 14:50:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 26A06

Related topic

SumOfSeries

Related topic

MultiplicationOfSeries

Generated on Fri Feb 9 21:15:13 2018 by

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My first Test

My first Test

My first Test

The solution to

x

=

5

is

x

=

25

.

Title

My first Test

Canonical

name

MyFirstTest1

Date of creation

2013-03-11 19:52:25

Last modified on

2013-03-11 19:52:25

Owner

jonmhays (21813)

Last modified by

(0)

Numerical id

1

Author

jonmhays (0)

Entry type

Definition

Generated on Fri Feb 9 21:15:19 2018 by

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non-existence of universal series convergence criterion

non-existence of universal series convergence criterion

There exist many criteria for examining the

convergence

and divergence of series with positive terms (see e.g.

determining series convergence

).  They all are

sufficient

but not necessary.  It has also been asked whether there would be any criterion which were both sufficient and necessary.  The famous mathematician

Niels Henrik Abel

took this question under consideration and proved the

Theorem

.

There is no

sequence

ϱ

1

,

ϱ

2

,

ϱ

3

,

…

(1)

of positive numbers such that every series

a

1

+

a

2

+

a

3

+

…

of positive terms

converges

when the condition

lim

n

→

∞

⁡

ϱ

n

⁢

a

n

=

0

is true but diverges when it is false.

Proof.

Let’s assume that there is a sequence (1) having the both properties.  We infer that the series

1

ϱ

1

+

1

ϱ

2

+

1

ϱ

3

+

…

is

divergent

because

lim

n

→

∞

⁡

(

ϱ

n

⋅

1

ϱ

n

)

≠

0

.  The

theorem

on

slower divergent series

guarantees us another

divergent series

s

1

+

s

2

+

s

3

+

…

such that the ratio

s

n

:

1

ϱ

n

=

ϱ

n

⁢

s

n

tends to the limit

0

as

n

→

∞

.  But this limit result concerning the series

s

1

+

s

2

+

s

3

+

…

should mean, according to our

assumption

, that the series is convergent.  The

contradiction

shows that the theorem holds.

References

1

Ernst Lindelöf:

Differentiali- ja integralilasku ja sen sovellutukset III.2.

Mercatorin Kirjapaino Osakeyhtiö. Helsinki (1940).

Title

non-existence of universal series convergence criterion

Canonical name

NonexistenceOfUniversalSeriesConvergenceCriterion

Date of creation

2013-03-22 15:08:33

Last modified on

2013-03-22 15:08:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Related topic

SlowerDivergentSeries

Related topic

SlowerConvergentSeries

Related topic

ErnstLindelof

Generated on Fri Feb 9 21:15:25 2018 by

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Nonlinear wave equation

Nonlinear wave equation

Nonlinear wave equation

A general nonlinear wave equation can be expressed as

(

N

⁢

W

⁢

E

)

⁢

{

u

t

⁢

t

-

u

x

⁢

x

=

f

⁢

(

x

,

t

,

u

)

in

⁢

(

0

,

π

)

×

ℝ

u

⁢

(

0

,

t

)

=

u

⁢

(

π

,

t

)

=

0

∀

t

∈

ℝ

where

f

⁢

(

x

,

t

,

s

)

is a

continuous

,

monotonic

increasing or decreasing (in

s

)
real function on

[

0

,

π

]

×

ℝ

2

and

2

⁢

π

-periodic in

t

.
Nontrivial

periodic solutions

are of

interest

in many applications. Existence of
multiple or even infinitely many periodic solutions to some special cases
of (NWE), e.g.,

f

⁢

(

x

,

t

,

s

)

is superlinear in

s

and/or independent of the time

t

, have been proved by many authors such as P. Rabinowitz, V. Benci, M. Berti, etc.

Title

Nonlinear wave equation

Canonical

name

NonlinearWaveEquation1

Date of creation

2013-03-11 19:28:45

Last modified on

2013-03-11 19:28:45

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

Generated on Fri Feb 9 21:15:27 2018 by

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Nonvariational systems

Nonvariational systems

Nonvariational systems

A typical nonvariational elliptic system has the form

(

N

⁢

V

)

⁢

{

-

Δ

⁢

u

=

f

⁢

(

x

;

u

,

v

)

,

x

∈

Ω

-

Δ

⁢

v

=

g

⁢

(

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,

f

⁢

(

x

;

u

,

v

)

,

g

⁢

(

x

;

u

,

v

)

∈

𝒞

1

⁢

(

Ω

¯

×

ℝ

2

;

ℝ

)

in the

variables

(

u

,

v

)

∈

ℝ

2

.
Here, we further assume that there exists no

function

G

⁢

(

x

;

u

,

v

)

with

∇

⁡

G

=

(

f

,

±

g

)

or

∇

⁡

G

=

(

g

,

f

)

. Under this

assumption

,
it is easy to see that problem (NV) is nonvariational.

Title

Nonvariational systems

Canonical

name

NonvariationalSystems1

Date of creation

2013-03-11 19:28:55

Last modified on

2013-03-11 19:28:55

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

Generated on Fri Feb 9 21:15:29 2018 by

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On the Residue Theorem

On the Residue Theorem

On the

Residue Theorem

Swapnil Sunil Jain
December 26, 2006

On the Residue Theorem

The Residue Theorem

If

γ

is a simply closed

contour

and f is

analytic

within the region

bounded

by

γ

except for some finite number of poles

z

0

,

z

1

,

…

,

z

n

then

∫

γ

f

⁢

(

z

)

⁢

𝑑

z

=

2

⁢

π

⁢

i

⁢

∑

k

=

0

n

R

⁢

e

⁢

s

z

=

z

k

⁢

f

⁢

(

z

)

where

R

⁢

e

⁢

s

z

=

z

k

⁢

f

⁢

(

z

)

is the reside of

f

⁢

(

z

)

at

z

k

.

Calculating Residues

The

Residue

of

f

⁢

(

z

)

at a particular pole

p

depends on the

characteristic

of the pole.

For a single pole

p

,

R

⁢

e

⁢

s

z

=

p

⁢

f

⁢

(

z

)

=

lim

z

→

p

⁡

[

(

z

-

p

)

⁢

f

⁢

(

z

)

]

For a double pole

p

,

R

⁢

e

⁢

s

z

=

p

⁢

f

⁢

(

z

)

=

lim

z

→

p

⁡

[

d

d

⁢

z

⁢

(

z

-

p

)

2

⁢

f

⁢

(

z

)

]

For a n-tuple pole

p

,

R

⁢

e

⁢

s

z

=

p

⁢

f

⁢

(

z

)

=

lim

z

→

p

⁡

[

1

(

n

-

1

)

!

⁢

d

(

n

-

1

)

d

⁢

z

(

n

-

1

)

⁢

(

z

-

p

)

n

⁢

f

⁢

(

z

)

]

Evaluation of Real-Valued Definite Integrals

We can use the Residue theorem to evaluate real-valued

definite integral

of the form

∫

0

2

⁢

π

f

⁢

(

sin

⁡

(

n

⁢

θ

)

,

cos

⁡

(

n

⁢

θ

)

)

⁢

𝑑

θ

(1)

If we let

z

=

e

i

⁢

θ

, then

d

⁢

z

d

⁢

θ

=

i

⁢

e

i

⁢

θ

=

i

⁢

z

which implies that

d

⁢

θ

=

d

⁢

z

i

⁢

z

. Then using the

identity

cos

⁡

(

n

⁢

θ

)

=

1

2

⁢

(

z

n

+

z

-

n

)

and

sin

⁡

(

n

⁢

θ

)

=

1

2

⁢

i

⁢

(

z

n

-

z

-

n

)

, we can re-write (1) as

∫

γ

g

⁢

(

z

)

⁢

d

⁢

z

i

⁢

z

(2)

where

g

⁢

(

z

)

=

f

⁢

(

1

2

⁢

i

⁢

(

z

n

-

z

-

n

)

,

1

2

⁢

(

z

n

+

z

-

n

)

)

and

γ

is a contour that traces the

unit circle

. Then, by the Residue theorem, (2) is equal to

2

⁢

π

⁢

i

⁢

∑

k

=

0

n

R

⁢

e

⁢

s

z

=

z

k

⁢

(

g

⁢

(

z

)

i

⁢

z

)

=

2

⁢

π

⁢

i

i

⁢

∑

k

=

0

n

R

⁢

e

⁢

s

z

=

z

k

⁢

(

g

⁢

(

z

)

z

)

=

2

⁢

π

⁢

∑

k

=

0

n

R

⁢

e

⁢

s

z

=

z

k

⁢

(

g

⁢

(

z

)

z

)

where

z

k

are the poles of

g

⁢

(

z

)

z

.

Title

On the Residue Theorem

Canonical

name

OnTheResidueTheorem1

Date of creation

2013-03-11 19:29:41

Last modified on

2013-03-11 19:29:41

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

Generated on Fri Feb 9 21:15:46 2018 by

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p test

p

test

The following is an immediate corollary of the

integral test

.

Corollary

(

p

-Test).

A series of the form

∑

n

=

1

∞

1

n

p

converges

if

p

>

1

and diverges if

p

≤

1

.

Proof.

The case

p

=

1

is well-known, for

∑

n

=

1

∞

1

n

is the

harmonic series

, which diverges (see this proof (

http://planetmath.org/ProofOfDivergenceOfHarmonicSreies

)). From now on, we assume

p

≠

1

(notice that one could also use the integral test to prove the case

p

=

1

). In order to apply the integral test, we need to calculate the following

improper integral

:

∫

1

∞

1

x

p

⁢

𝑑

x

=

lim

n

→

∞

⁡

[

x

1

-

p

1

-

p

]

1

n

=

lim

n

→

∞

⁡

n

-

p

+

1

1

-

p

-

1

1

-

p

.

Since

lim

n

→

∞

⁡

n

t

diverges when

t

>

0

and converges for

t

≤

0

, the

integral

above converges for

1

-

p

<

0

, i.e. for

p

>

1

and diverges for

p

<

1

(and also diverges for

p

=

1

). Therefore, the corollary follows by the integral test.
∎

Title

p

test

Canonical name

PTest

Date of creation

2013-03-22 15:08:51

Last modified on

2013-03-22 15:08:51

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Corollary

Classification

msc 40A05

Synonym

p-test

Synonym

p

-test

Synonym

p test

Synonym

p series test

Synonym

p

-series test

Synonym

p

series test

Related topic

ExamplesUsingComparisonTestWithoutLimit

Related topic

ASeriesRelatedToHarmonicSeries

Generated on Fri Feb 9 21:15:56 2018 by

LaTeXML
proof of Abel lemma (by expansion)

proof of Abel lemma (by expansion)

1

Abel lemma

∑

i

=

0

n

a

i

⁢

b

i

=

∑

i

=

0

n

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

n

⁢

b

n

,

(1)

where,

A

i

=

∑

k

=

0

i

a

k

. Sequences

{

a

i

}

,

{

b

i

}

,

i

=

0

,

…

,

n

, are real or complex one.

2

Proof

We consider the expansion of the sum

∑

i

=

0

n

A

i

⁢

(

b

i

-

b

i

+

1

)

on two different forms, namely:

1.

On the short way.

∑

i

=

0

n

A

i

⁢

(

b

i

-

b

i

+

1

)

=

∑

i

=

0

n

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

n

⁢

b

n

-

A

n

⁢

b

n

+

1

.

(2)

2.

On the long way.

∑

i

=

0

n

A

i

⁢

(

b

i

-

b

i

+

1

)

=

∑

i

=

0

n

A

i

⁢

b

i

-

∑

i

=

0

n

A

i

⁢

b

i

+

1

=

∑

i

=

0

n

A

i

⁢

b

i

-

∑

i

=

1

n

+

1

A

i

-

1

⁢

b

i

=

A

0

⁢

b

0

+

∑

i

=

1

n

(

A

i

-

1

+

a

i

)

⁢

b

i

-

∑

i

=

1

n

A

i

-

1

⁢

b

i

-

A

n

⁢

b

n

+

1

=

∑

i

=

0

n

a

i

⁢

b

i

-

A

n

⁢

b

n

+

1

,

(3)

where a simplification has been performed. Notice that

A

0

=

a

0

. By equating (2), (3), the last two terms cancel,

1

1

Without loss of generality,

b

n

+

1

may be assumed finite. Indeed we don’t need

b

n

+

1

, but the proof is a couple lines larger. It is left as an exercise.

and then, (1) follows.

□

Title

proof of Abel lemma (by expansion)

Canonical name

ProofOfAbelLemmabyExpansion

Date of creation

2013-03-22 17:28:14

Last modified on

2013-03-22 17:28:14

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

7

Author

perucho (2192)

Entry type

Proof

Classification

msc 40A05

Generated on Fri Feb 9 21:16:36 2018 by

LaTeXML
proof of Abel’s lemma (by induction)

proof of Abel’s lemma (by induction)

Proof.

The proof is by

induction

. However, let us
first recall that sum on the right side is a
piece-wise defined

function

of the

upper limit

N

-

1

.
In other words, if the upper limit is below the

lower
limit

0

, the sum is identically set to zero.
Otherwise, it is an ordinary sum.
We therefore need to manually check the first two cases.
For the

trivial

case

N

=

0

, both sides equal to

a

0

⁢

b

0

.
Also, for

N

=

1

(when the sum is a normal sum), it is easy to verify that
both sides simplify to

a

0

⁢

b

0

+

a

1

⁢

b

1

.
Then, for the induction step, suppose that the
claim holds for some

N

≥

1

. For

N

+

1

, we then have

∑

i

=

0

N

+

1

a

i

⁢

b

i

=

∑

i

=

0

N

a

i

⁢

b

i

+

a

N

+

1

⁢

b

N

+

1

=

∑

i

=

0

N

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

N

⁢

b

N

+

a

N

+

1

⁢

b

N

+

1

=

∑

i

=

0

N

A

i

⁢

(

b

i

-

b

i

+

1

)

-

A

N

⁢

(

b

N

-

b

N

+

1

)

+

A

N

⁢

b

N

+

a

N

+

1

⁢

b

N

+

1

.

Since

-

A

N

⁢

(

b

N

-

b

N

+

1

)

+

A

N

⁢

b

N

+

a

N

+

1

⁢

b

N

+

1

=

A

N

+

1

⁢

b

N

+

1

,
the claim follows.

□

.

Title

proof of Abel’s lemma (by induction)

Canonical name

ProofOfAbelsLemmabyInduction

Date of creation

2013-03-22 13:38:04

Last modified on

2013-03-22 13:38:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Proof

Classification

msc 40A05

Generated on Fri Feb 9 21:16:38 2018 by

LaTeXML
proof of Abel’s test for convergence

proof of Abel’s test for convergence

Let

b

be the limit of

{

b

n

}

and let

d

n

=

b

n

-

b

when

{

b

n

}

is decreasing and

d

n

=

b

-

b

n

when

{

b

n

}

is increasing. By Dirichlet’s

convergence test

,

∑

a

n

⁢

d

n

is

convergent

and so is

∑

a

n

⁢

b

n

=

∑

a

n

⁢

(

b

±

d

n

)

=

b

⁢

∑

a

n

±

∑

a

n

⁢

d

n

.

Title

proof of Abel’s test for

convergence

Canonical name

ProofOfAbelsTestForConvergence

Date of creation

2013-03-22 13:19:56

Last modified on

2013-03-22 13:19:56

Owner

lieven (1075)

Last modified by

lieven (1075)

Numerical id

4

Author

lieven (1075)

Entry type

Proof

Classification

msc 40A05

Generated on Fri Feb 9 21:16:40 2018 by

LaTeXML
proof of absolute convergence theorem

proof of absolute convergence theorem

Suppose that

∑

a

n

is

absolutely convergent

, i.e., that

∑

|

a

n

|

is convergent.
First of all, notice that

0

≤

a

n

+

|

a

n

|

≤

2

⁢

|

a

n

|

,

and since the series

∑

(

a

n

+

|

a

n

|

)

has non-negative terms it can be compared with

∑

2

⁢

|

a

n

|

=

2

⁢

∑

|

a

n

|

and hence

converges

.

On the other hand

∑

n

=

1

N

a

n

=

∑

n

=

1

N

(

a

n

+

|

a

n

|

)

-

∑

n

=

1

N

|

a

n

|

.

Since both the

partial sums

on the right hand side are convergent, the partial
sum on the left hand side is also convergent. So, the series

∑

a

n

is convergent.

Title

proof of absolute convergence theorem

Canonical name

ProofOfAbsoluteConvergenceTheorem

Date of creation

2013-03-22 13:41:52

Last modified on

2013-03-22 13:41:52

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

6

Author

paolini (1187)

Entry type

Proof

Classification

msc 40A05

Generated on Fri Feb 9 21:16:43 2018 by

LaTeXML
proof of alternating series test

proof of alternating series test

The series has

partial sum

S

2

⁢

n

+

2

=

a

1

-

a

2

+

a

3

-

+

…

-

a

2

⁢

n

+

a

2

⁢

n

+

1

-

a

2

⁢

n

+

2

,

where the

a

j

’s are all nonnegative and

nonincreasing

.
From above, we have the following:

S

2

⁢

n

+

1

=

S

2

⁢

n

+

a

2

⁢

n

+

1

;

S

2

⁢

n

+

2

=

S

2

⁢

n

+

(

a

2

⁢

n

+

1

-

a

2

⁢

n

+

2

)

;

S

2

⁢

n

+

3

=

S

2

⁢

n

+

1

-

(

a

2

⁢

n

+

2

-

a

2

⁢

n

+

3

)

=

S

2

⁢

n

+

2

+

a

2

⁢

n

+

3

Since

a

2

⁢

n

+

1

≥

a

2

⁢

n

+

2

≥

a

2

⁢

n

+

3

, we have

S

2

⁢

n

+

1

≥

S

2

⁢

n

+

3

≥

S

2

⁢

n

+

2

≥

S

2

⁢

n

. Moreover,

S

2

⁢

n

+

2

=

a

1

-

(

a

2

-

a

3

)

-

(

a

4

-

a

5

)

-

⋯

-

(

a

2

⁢

n

-

a

2

⁢

n

+

1

)

-

a

2

⁢

n

+

2

.

Because the

a

j

’s are nonincreasing, we have

S

n

≥

0

for any

n

. Also,

S

2

⁢

n

+

2

≤

S

2

⁢

n

+

1

≤

a

1

. Thus,

a

1

≥

S

2

⁢

n

+

1

≥

S

2

⁢

n

+

3

≥

S

2

⁢

n

+

2

≥

S

2

⁢

n

≥

0

. Hence, the even partial sums

S

2

⁢

n

and the odd partial sums

S

2

⁢

n

+

1

are bounded. Also, the even partial sums

S

2

⁢

n

’s are

monotonically nondecreasing

, while the odd partial sums

S

2

⁢

n

+

1

’s are monotonically nonincreasing. Thus, the even and odd series both

converge

.

We note that

S

2

⁢

n

+

1

-

S

2

⁢

n

=

a

2

⁢

n

+

1

. Therefore, the sums converge to the

same

limit if and only if

a

n

→

0

as

n

→

∞

. The theorem is then established.

Title

proof of alternating series test

Canonical name

ProofOfAlternatingSeriesTest

Date of creation

2014-07-22 16:20:39

Last modified on

2014-07-22 16:20:39

Owner

Wkbj79 (1863)

Last modified by

pahio (2872)

Numerical id

13

Author

Wkbj79 (2872)

Entry type

Proof

Classification

msc 40A05

Generated on Fri Feb 9 21:16:45 2018 by

LaTeXML