subsidies for a generic theory about equivalences

8/14/2019 Subsidies for a Generic Theory About Equivalences

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SUBSIDIES FOR A GENERIC THEORY ABOUT EQUIVALENCES BETWEEN

NUMERICAL SERIES AND SUCCESSIONS

Introduction

The study ofNumericalSeries ( )NS understood as the sum

( )of the infinite terms

of successions here designated by base successions ( )nb or, symbolically,

( )

=

=1n

nbNS has been until today, mainly, consisting into the clarification about its

convergence or divergence.

However, ideally, the referred study should be the rigorous determination and

preferentially using simple processes, of the sum of any number of terms of nb

including thesum of its infinite terms ( )SIT (or limit of nb sum) if such sum existsdefining a tendency real numerical value.

In fact, it is always possible the determination of sum of any number of terms of nb

doing it directly.

However, the calculation of that direct sum is very fastidious and more and more

complicated when the number of summing terms augment.

And still worse, the attempt of SITcalculation, using the direct sum of nb terms is,

absolutely, an impossible task.

Then, the indicated ideal study of aNSonly may have success using indirect methods of

calculation of referred sums.

In some (few) types ofNS,those indirect methods are effective, using the substitution of

the givenNSby an equivalentassociatedsuccessions ( )nS establishing equivalences of

type ( )

=

=1n

nn SbNS .

In those equivalences, the successive numerical values of nS does the substitution of

direct calculation of the sum of any finite number of nb terms, including the SIT

numerical value, when it exists.

In fact, for valid equivalences nSNS , we can assure that:

The first term ofNS is ( ) ( )=

=

===1

1

111

n

n

n SbbNS

The second term ofNS is ( ) ( )==

=+==2

1

2212

n

n

n SbbbNS

The third term ofNS is ( ) ( )=

=

=++==3

1

33213

n

n

n SbbbbNS

The term of n order of NS, or generic term, will be represented by

( ) ( )=

=

=++++==nn

n

nnnn SbbbbbNS1

321 ...

Finally, the SITnumerical value of theNSwill be:

( ) ( )

=+ =++++++==

1

121 lim......n

nnnnSbbbbbbSIT

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Note: We understandb as being ( )nblim

This article shows that we can amplify the present knowledge about equivalences

nSNS in a significant manner.

The starting pointThe present situation about the study ofNS is based into the utilization of severalConvergenceCriteria defining the convergence or divergence of theNS.

The determination of equivalences nSNS is, normally exceptional, only beingpossible for a few types ofNS.

Additionally, to some otherNS, although it is not possible define complete equivalences

nSNS we may use special methods for determination of respective SITonly.Thus, according with present knowledge we have:

i) TheArithmetic Series (AS) whose banbn += { }( )IbI Ra ,0\are called

Arithmetic Progressions (AP), allow equivalences nSNS . Its generic

expression is ( )

=

+

==1

1

2n

nnn n

bbSbAS . Obviously, the SITnumerical

value of allASare always infinite (positive or negative).

ii) The Geometric Series (GS) whosen

nrb = { }( )0\I Rr are called Geometric

Progressions (GP) and allow, as well, equivalences nSNS . Its generic

expression is ( )=

==1 1

1

n

n

n

n

rrrSrGS . The GSare convergent only for

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The creation of new equivalences nSNS is the objective of the first part of thisarticle. Then, we propose two different methods of creation of exact equivalences

nSNS which are the following:i) First method : Starting from a given NS and writing its successive terms

equals to progressive sums of nb terms, search detect standard behaviours

of the corresponding sequent partial sums which can lead us to understand

the equations of respective exact equivalence nSNS .ii) Second method : Starting from several successions nu , all belonging to same

functional group, search establish specific relationships nnn Sbu usingvery simple and understandable fundaments described next. Analysing then

the several results obtained, search detect standard behaviours which give

support to the creation of formulas and processes able to define exact

equivalences nSNS to the referred functional groups.

1.2. First method to search exact equivalences nSNS

As we said before, the first method to search exact equivalences nSSN is based intowriting successive terms of one chosenNSin order to detect standard behaviours which

lead us to establish an exact equivalence nSNS .

Example: Consider the

=

=

1 !

1

n n

nNS

Let us determine the first four successive terms ofNS. We obtain:

The first term ( )1NS will be ( ) =

=

===

=

1

1

110

!1

11

!

1n

n

bn

nNS

The second term ( ) 2NS will be ( ) ==

=+=+=

=

2

1

21221

!2120

!1

n

n

bbn

nNS

The third term is ( ) =

=

=

+=+=++=

=

3

1

3232136

5

!3

13

2

1

!

1n

n

bSbbbn

nNS

The fourth term is ( ) =

=

=+=+=

=

4

1

43424

23

8

1

6

5

!

1n

n

bSn

nNS

The numerical values obtained to the four sequential terms of theNSmay be written as:

( )!0

1!001

==NS ; ( )

!2

1!2

2

12

==NS ; ( )

!3

1!3

6

53

==NS ; ( )

!4

1!4

24

234

==NS .

Since those four results, seems that we may risk write

=

=

=

1 !

1!

!

1

n

nn

nS

n

nNS

In fact, that equivalence is valid (as we demonstrate later) and as we can confirm

provisionally right now, determining the equality for more two terms. Thus to 4>n :

The fifth term will be ( ) =

=

==

+=

=

5

1

5!5

1!5

120

119

!5

15

24

23

!

1n

n n

nNS

The sixth term will be ( ) =

=

==

+=

=

6

1

6!6

1!6

720

719

!6

16

120

119

!

1n

n n

nNS

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Note: The described first method proposed for determination of exact equivalences nSNS may beused to others NS. However, it is normally very laborious, fallible many times and hardly allows

generalizations. Thus, its interest is restrained.

1.3. Second method to search exact equivalences nSNS

1.3.1. Introduction. Presentation of Method.Equivalence Principle.

The second method to search exact equivalences nSNS is much more advantageousand it is based into the following fundaments.

The nS succession exactly equivalent to aNSis a normal succession, similar to anyother succession nu , given by a functional equation with natural variable like

( )nfSn = .

However, any nS have, relatively to a normal succession nu , to respond to anadditional requisite which impose that the sequence nSSSS ,...,,, 321 of its terms must

represent the progressive sum of the terms of another succession nb as being:

11 bS =

212 bbS +=

3213 bbbS ++=

13211 ... ++++= nn bbbbS

nn bbbbS ++++= ...321

Subtracting the two previous last equations 1 nn SS we obtain:

( ) nnnnnnnn bSSbbbbbbbbbSS =+++++++++= 1132113211 ......

Based into generic equation nnn bSS = 1 we can enunciate the useful

Equivalence Principle nSNS ( nSNSEP , )

If a NS with succession base nb has an associated equivalent succession nS orif ( )

=

=1n

nn SbNS then nnn bSS = 1 and nn bbbbS ++++= ...321

Example: Lets use the nSNSEP , to confirm the previous exact equivalence

=

=

=

1 !

1!

!

1

n

nn

nS

n

nNS .

We have( )

( ) !1

!1

1!1

!

1!1

n

nb

n

n

n

nbSSb nnnnn

=

== .

The result of nb proves the validity of equivalence.

Note: The nSNSEP , is referred and described in many books and texts about NS. However, thenew approach made to it lead us to developments very interesting and never explored before.

1.3.2. The possibility of any succession nu may generate an exact equivalence

nSNS

Any succession nu is always able to generate another succession nv through the

operation nnn vuu = 1 .

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On other hand, it is obvious that the equations nnn vuu = 1 and nnn bSS = 1 aresimilar.

Then, we may ask the interrogation. If nnnn Subv == ?In fact, that situation may happen (and with remarkable frequency) but, unfortunately,

not ever, as we will see next.

1.3.3. Preliminary study of the utilization of nSNSEP , in order to lead onesuccession nu to generate a valid exact equivalence nSNS .When, we choose a succession nu , in order to use it to generate a valid exact

equivalence nSNS , we start doing the difference nnn vuu = 1 hoping that if

nnbv = then ( )

=

==1n

nnn uSbNS .

The validity of that equivalence imposes the equality nn uS = .However, the equality nn uS = is not the only possibility of nu may generate an exact

equivalence nSNS .In fact, if we consider CuSCuS nnnn +=+= 11 with IRC then we can writealso ( ) nnnnnnnnn buubCuCubSS ==++= 111 .

Thus, the equivalence ( )

=

+==1n

nnn CuSbNS will be a more all-embracing

possibility of nu may generate an exact equivalence nSNS .The new exact equivalence includes now an unknown constant C. Then, it is necessarycalculate the Cnumerical value. That calculationis very easy as we see next.

In fact, if 1111 uSCCuSCuS nn =+=+=

Then we may write ( ) ( )

=

=+==+==

1

11

1n n

nnnnnn uSuSbNSCuSbNS

Unfortunately, the final results for the exact equivalences obtained by this method are

not always absolutely secure.

So, it will be advised consider any final equivalence nSNS coming from thismethod as only provisional until we can confirm the final expression.

First example: Consider the exponential succession nn ru = with { }0\I Rr .Lets determine the corresponding nb using the nSNSEP , .

We obtain ( )1111 ===

rrbrrbuubn

n

nn

nnnn .

Then, if CuS nn += we can write ( )[ ]

=

+==1

11

n

n

n

nCrSrrNS .

Calculation of C numerical valueWe have 1111 === CrrCubC .Thus, the provisional exact equivalence nSNS to the example in study will be:

( )[ ] ( ) ( )

=

=

=

==

=1 1 1

1

1

11

111

n n n

n

n

nn

n

nn

n

n

r

rrSrrSr

r

rrSrr No

te:The obtained provisional exact equivalence ( )

=

== 1 11

n

n

n

n

r

r

rSrNS is recognizing valid

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because it is the definitively known equivalence nSNS applicable to the Geometrical Seriesformerly presented in this text and enough demonstrated.

Second example: Consider the rational succession( ) 21+

=n

nu

n

Lets determine ( ) ( ) 22

2

2211

11

1 +++

=

+== nnnn

bn

n

n

n

buub nnnnn

Then we may write( ) ( )

=

++

=

+++

=1

222

2

11

1

n

n Cn

nS

nn

nnNS

Calculation of C numerical valueWe have nn SuCubC ==== 0

4

1

4

111

Thus, we may write the provisionally exact equivalence as being:

( ) ( ) 21 22

2

11

1

+=

+

++=

= n

nS

nn

nnSN n

n

Note: The obtained provisionally exact equivalence is obviously invalid because nb and nS are

incompatible since nb is a negative succession while nS is a positive succession.

This situation of invalidity for provisionally exact equivalences coming from the nSNSEP , mayoccur to other types of nu successions.

So, to prevent those failures, the ideal should be identify, a priori, if nu has the capacity to generate

valid equivalences.

Next we will speak about that matter.

2. Necessary and sufficient conditions to one succession nu can generate a validexact equivalence nSNS

2.1. Necessary but not sufficient conditions

2.1.1. Introduction. Incorporation of useful symbology

Before starting the study of conditions in title lets introduce some new symbology in

order to simplify the sequent text.

This new symbology consists into divide all the successions into four great groups (or

conjuncts) attributing it a designation and a symbol according its nature or limit.

Thus, we will consider the successions as being:

i) Successions Infinitely Great( )SIG , when they have infinite limit.ii) Successions Infinitely Small( )SIS , when they have limit zero.

iii) Convergent Successions Except SIS( )S IC S\ , when they have finite anddeterminate real limit which can be any real number different of zero.

iv) Convergent Successions ( )CS , when they have real limit (including zero).v) Successions Without Defined Limit ( )SWDL , when they do not have an

unique and defined real limit.

Note: According the proposed symbology, writing for example, S IS Cun

\ means that nu belongs to

the conjunct of convergent successions but with limit different of zero.

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On other hand, writing SISun means that nu belongs to the conjunct of successions infinitelysmall, characterized by having specific limit zero.

2.1.2. Initial considerations

If an exact equivalence nSNS obtainedthrough the application of the nSNSEP ,

to a succession nu is valid, because it is already confirmed or because it is susceptibleof confirmation, then the characteristics of the intervenient successions nu , nb and

nS must have inter-conditionings related with signal, nature or limit and yet with

evolutivebehaviour of terms. Lets analyse it next. Thus:

2.1.3. Conditioning of signal

Obviously, any valid exact equivalence coming from nSNSEP , impose that theinvolved successions 1= nnn uub and nn bbbS +++= ...21 must have the same

signal.

That condition excludes immediately the successions nu without fix signal as being

able to establish those equivalences.Also the SWDLun are not able to establish equivalences nSNS .

Example: The succession( ) 21+

=n

nu

n , previously referred, originate an invalid

equivalence because, as we have seen, nb and nS had incompatible signal.

Note: The conditioning of signal seems indicate that successions SISun (as the succession of formerexample) are not able to obtain valid exact equivalences nSNS .However, it is not so. Lets see the following example.

Example: Consider the succession SISnun += 112

Lets determine the correspondent( ) ( ) ( )221

12

11

1

1

12222 +++

=+

+

=nnn

nb

nnb

nn

We see that nb has negative signal while nu has positive signal which seems to be incompatible.

However, we may write( ) ( )

=

++

=

++

+=1

2221

1

221

12

n

n Cn

Snnn

nNS

Calculation of C numerical value

We have 12

1

2

111 === CubC . Then:

( ) ( )

= +=

++

+=+

=+

=1

2

2

222

2

21221

12

11

1

1

n

nnnn

nS

nnn

nSN

n

nS

nS

As we see the C numerical value conciliates the signal between nb and nS fixing it into negative

signal for both successions nb and nS .

Additionally, we may advance that the obtained exact equivalence is correct.

2.1.4. Conditionings of nature or limit.

According the theory ofNS, the nature of nb imposes the nature of SN (or )nS .

On other hand, the nature of nS imposes the nature of the initial nu used into

nSNSEP , in following manner.

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i) If SuS I SC SuS I SC SSS I Sbnnnn

\\ (as we explain before, the

SISun imposes a changing of signal for nS )

ii) If S IuS I GSS I SS Cb nnn \iii) If SIGuSIGSSIGb nnn

Example: Consider the succession nun = .The considered succession is SIGun .

According the nSNSEP , we may write 1= nnn uub . Thus:( ) ( )

1

1

1

1111

++=

++++

=== nn

bnn

nnnnbnnbuub nnnnnn

We may write

=

+=

++=

1 1

1

n

n CnS

nn

NS

Calculation of C numerical valueAs we know nn uSCCubC ==== 01111

Then we can write provisionally

=

=

++

=1 1

1

n

n nSnn

NS

Then, if the former provisionally equivalence is valid, in that case, the relation of

natures is SIGuSIGSSISb nnn . Next we demonstrate the validity for that.

Important Note: Demonstration of validity of provisionally exact equivalences. Method of successiveverifications

Having in consideration the risks of invalidity of any exact equivalence coming from the

nSNSEP , it is necessary verifying those validities.To get that, we may use the Method of Successive Verifications which consists into making the

verification of equality of successive terms of SN and nS

Since the calculation of C put 11 SNS = , then it is enough verify the equality of second term andterm of n order.If those equalities are verified then we can assure the validity of the equivalence nSNS in analyse.

Example: Verification of the provisionally equivalence previously obtainedApplying the Method of Successive Verifications to 2=n and nn = terms we have:

To 212

1122 2122 =

++=+=== bbNSSn (confirms)

To nnn

nbSNSnSnn nnnn =+

+=+=== 1

111 (confirms)

Thus, according the Method of Successive Verifications we can assure that the exact equivalence obtainedinto previous example is absolutely valid.

Remark: The Simplified Method of Successive Verifications.

The Method of Successive Verifications may be used to verify any exact equivalences nSNS in a

completely warrantable manner.However, its application should not always easy, mainly the verification of term nnn bSS += 1 .

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So, many times, we can use the Simplified Method of Successive Verifications which consists into making

only the two first successive verifications after the calculation ofCnumerical value ( )2>n .In fact, if the equivalence is not correct, one error will be detected for the prime verification and

confirmed for the second verification.

Thus, the Simplified Method of Successive Verifications is also completely warrantable.

Example: Lets use the Simplified Method of Successive Verification to demonstrate the formerequivalence

( ) ( )

= +

++

+=

12

2

22 1221

12

n n

n

nnn

nNS . Thus:

To5

4

5

4

10

3

2

12 2212 ===+== SbbNSn (confirms)

To10

9

10

9

10

1

5

43 3323 ===+== SbSNSn (confirms)

2.1.5. Conditionings of evolutive behaviour of the terms.

Firstly we will refer only the conditionings of evolutive behaviour of terms concerning

relationships nn Sb with positive signal.To the same successions with negative signal the conditionings may be adjusted having

in account the symmetry of behaviours. Thus:

i) The SISbn with positive terms monotonously decrescent may originate

S IC SSn

\ or SIGSn with positive terms monotonously crescents.

ii) The S IS Cbn

\ with positive terms monotonously crescents or monotonouslydecrescent to own limit will originate SIGSn with positive terms

monotonously crescents.iii) The SIGbn with positive terms monotonously crescents will originate

SIGSn with positive terms monotonously crescents.

Note: The valid exact equivalence

=

=

+

=1 1

1

n

n nSnn

NS obtained behind is composed

by a SIPbn with positive terms monotonously decrescent to zero, originating a SIGSn withpositive terms crescent to infinite.

2.1.6. Resume of conditionings. Viable successions.

Resuming the conditionings explained until now, we can stand that when we are in

presence of any succession nu which we want use to try obtain a valid exactequivalence nSNS , using the nSNSEP , , we should verify (a priori) if nu andthe corresponding 1= nnn uub obey to the following conditionings.Thus, the succession nu must:

i) Have terms with fix signal (positive or negative).

ii) If S IC Sun

\ or SIGun have positive signal, its terms must bemonotonously crescents.

iii) If SISun have positive signal, then its terms must be monotonously

decrescent.

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iv) If S IC Sun

\ or SIGun have negative signal, its terms must bemonotonously decrescent.

v) If SISun have negative signal then its terms must be monotonouslycrescents.

vi) nu must generate 1= nnn uub with the following characteristics:1) Have terms with same fix signal of nu except if SISun .2) Have terms monotonously decrescent if SISbn with positive signal or

monotonously crescents if SISbn with negative signal.

3) Have terms monotonously crescents if S IC Sbn

\ or SIGbn with positive

signal, or monotonously decrescent if S IC Sbn

\ or SIGbn with negativesignal.

Note: To economize words in following text we designate, since now, as viable successions all the

successions nu or nb which obey the necessary conditionings to be able to generate valid exact

equivalences.

2.1.7. Examples of analyse of viable successions nu and determination of

corresponding exact equivalences nSNS when possible.

First example: Consider the succession1

22

2

++

=n

nnun

It is a viable succession? Lets see. The succession S IS Cun

\ and its four first terms are,

17

24;2

3;5

8;2

34321 ==== uuuu . On other hand, 1

1lim

2

2

=++

n

nn.

The numerical values for the four first terms of nu indicate that+1nu but not

monotonously. Then, nu is not a viable succession.

Second example: Consider the succession,( ) 3

3

1

1

++

=n

nun . It will be viable?

The S IC Sun

\ has positive signal, its ( ) =1lim nu and it is monotonously crescent to itsown limit. Then, it could be, in principle, a viable succession.

Lets determine the corresponding nb .

We obtain( )

( )

( ) 33

24

3

3

3

3

1

36311

1

1

+

=+

++

=nn

nnnb

n

n

n

nb nn .

The succession SISbn . Its four first terms are400

33;

86

9;

12

1;

4

34321 ==== bbbb .

The non monotony of the decreasing to zero of nb and also the fact of the prime term is

negative, leads us to conclude that nb is not a viable succession.

Consequently, we have to reject the initial nu to be able to generate a valid exact

equivalence nSNS .

Third example: Consider the succession ( ) 22

1+= nn

un . It is viable?

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The S IC Sun

\ has positive signal, its ( ) =1lim nu and it is monotonously decrescent toown limit.

So, it could be, in principle, a viable succession. Let us determine the corresponding nb

.

We have( )

( )

( ) 22

2

2

2

2

2

11

121

1 +=

+==

nn

nb

n

n

n

nbuub nnnnn .

The SISbn , it has positive signal and 0

nb monotonously.

Thus, nu still may be viable.

Then, we may write( ) ( )

Cn

nS

nn

nNS

n

n ++=

+

=

=12

2

22

2

11

12


As we know ( ) 2

2

111004

1

4

1

+====== nn

SuSCCubC nnn

We may write provisionally( ) ( )

= +=

+

=1

2

2

22

2

11

12

n

nn

nS

nn

nNS

Simplified Method of Successive Verifications To

9

4

36

7

4

1

9

42

2122 =+=+=== bbNSSn (confirms)

To16

9

16

93 3233 =+=== bSNSSn (confirms)

As final conclusion, we can say that the former succession nu was a viable succession

and it has generate a valid exact equivalence nSNS .

Note:Although the nSNSEP , is very useful to obtain important exact equivalences nSNS since initial viable successions nu , truly, it is fallible.

So, the ideal should be dispense those initial successions nu and search establish valid exact

equivalences nSNS directly since viable successions considered as nb .So, the following considerations intend to respond to this objective.

3. Necessary and sufficient conditions to any viable succession considered as nb

may generate, directly, valid equivalences nSNS

3.1. Introduction. Notion ofadequate decomposition of nb

As we demonstrate before, all the exact equivalences ( ) nn

n SbNS =

=1will be valid if

respect both conditions (1) 1= nnn SSb and (2) nn bbbbS ++++= ...321 .Thus if, since a succession considered nb , we can write the equation 1= nnn SSb

being nn bbbbS ++++= ...321 then we can stand the validity of ( )

=

=1n

nn SbNS .

Then, the decomposition 1= nnn SSb with nn bbbS +++= ...21 will be thenecessary

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and sufficient conditions to establish a valid exact equivalence nSSN .If that decomposition 1= nnn SSb is possible we call it the adequate decompositionof nb . Then, starting from a certain succession considered nb and to search the

equivalence nSSN we intent obtain the adequatedecomposition 1= nn SSb .

Example 1: Consider the succession ( )11

+= nnbn

Lets search the normal decomposition.

We obtain( ) ( ) 1

11

1

1

11

1

+=

+

++=

+ nnnnnB

n

A

nn.

The obtained decomposition was of type nnn uub = 1 considering1

1

+=

nun .

That decomposition is not de adequatedecomposition 1= nnn SSb and still worst wedo not know if nn Su =

In spite of that we may write

( )

=

+

+

=

+

=1 1

1

1

1

n

nC

n

S

nn

NS

Calculation of CWe obtain 1

2

1

2

111 =+== CCubC

Then( ) ( )

=

= +=

+

=++

=

+

=1 1 11

11

1

1

1

1

n n

nnn

nS

nnNS

nS

nnNS

Simplified method of verification To

3

2

6

1

2

1

3

22 2122 =+=+=== bbNSSn (confirm)

To4

3

12

1

3

2

4

33

3233 =+=+=== bSNSSn (confirm)

Note: The obtained equivalence is valid

Example 2: Consider the same succession ( )11

+=

nnbn

Lets search a new decomposition imposing 1= nnn uub .

We obtain( )

( )( ) n

n

n

n

nnn

nA

n

An

nn

1

11

11

11

1 +

=+

+

=+

The obtained decomposition is of type 1= nnn uub but we do not know if nn Su = .

However we may write ( )

= ++=

+= 1 11

1

n

n Cn

n

SnnNS

Calculation of CWe have 0

2

1

2

11

=== CCubC

Then

= +=

+=

1 11

1

n

nn

nS

nnNS which is the same result obtained behind and

already verified.

Thus, the decomposition( ) n

n

n

n

nn

1

11

1

+=

+ is the adequate decomposition

1= nnn SSb and 1+= nn

Sn

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Note: Both decompositions nnn uub = 1 or 1= nnn uub can be utilized to obtain the exactequivalence nSNS

Then we are in conditions of enunciate the

Theorem of Adequate Decomposition (TAD)

Given a viable succession nb that can be decomposed into the subtraction of two sequent generic terms of another succession nu either nnn uub = 1 or

1= nnn uub then, to any of referred decompositions it exists a constant IRC such as, defining respectively CuS nn += or CuS nn += , will change it into theadequate decomposition 1= nnn SSb since we can write, at once, the valid exact

equivalence ( )

=

=1n

nn SbNS which imply nn bbbbS ++++= ...321

Note: Obviously the initial perception that a viable succession nb can not be decomposed according the

TAD take us to reject it, immediately, as a succession able to generate a valid exact equivalence

nSNS .

Example: Consider the succession34

12 ++

=nn

bn . The succession SISbn is viable.

However, the denominator of nb such as ( ) ( )31342 ++=++ nnnn is the product of two not

sequent terms of a polynomial succession which becomes invalid the possibility of any adequate

decomposition of nb .

So, in that case, the succession nb , in spite of to be viable, can not generate a valid exact equivalence

nSNS because it does not respect the TAD.

We are now in conditions of searching systematized methods for determination of valid

equivalences nSNS since viable nb successions.

4. Creation of systematized methods for determination of valid exact equivalences

nSNS since viable successions nb belonging to a chosen functional group.4.1. Introduction.

Recurring to the nSNSEP , , since several viable successions nb belonging to onesame functional group, we may create systematizations which allow the definition of

generic formulas in order to obtain, directly, valid exact equivalences nSNS .

Then we begin by the polynomial successions.

4.2. Polynomial successions

4.2.1. Introduction

The polynomial successions have generic equation

mm

mm

nanananau ++++=

1

1

10 ...

being mm aaaaa ,,...,,, 1210 real coefficients constants and 0Nm will be theexponent with greater order designated by degree of the polynomial succession.

The polynomial successions are always SIGun except those with 0=m which willbe 0

0

0 aunau nn == having constant terms.

13


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The following study consists into choosing some polynomial successions nu with

positive terms monotonously crescents for infinite and, using the nSNSEP , ,analyse the standard behaviours of the resulting valid exact equivalences nSNS .

*Note: As the extensive representation of generic polynomial successions is not practical for the frequent

writing used forwards, we will pass to use simplified symbology like ( ) ( )nPu mn = or ( ) ( )nQv pn = ,

or yet( ) ( )nRw qn = as representing polynomial successions with variable Nn and degrees

( ) Nqpm ,, respectively.Thus, if we write

( ) ( )14 = nQun we are saying that it is a polynomial succession with degree 4 andvariable ( )1n .

4.2.2. Study of valid equivalences resulting of polynomial successions nu .

We will present the study of only two significant examples.

First example: Consider( )

( )22

nunPu nn==

.We have ( ) 121

22

1 === nbnnbuub nnnnn

Calculation of C numerical valueAs 01111 === CubC

We can write the provisionally exact equivalence ( )

=

==1

212

n

n nSnNS

Note:The obtained result for the equivalence is correct because it is already known and confirmed by thegeneric formula applied to arithmetic series.

Second example: Consider( )

( ) nnunPu nn ==33

.

The corresponding ( ) ( ) nnbnnnnbuub nnnnn 3311233

1===

Calculation of C numerical valueAs 00011 === CubC .

We can write the provisionally exact equivalence ( ) nnSnnNS nn

==

=

3

1

233

Simplified Method of Successive Verifications To 66062 2122 =+=+=== bbNSSn (confirms)

To 24186243 3233 =+=+=== bSNSSn (confirms)

Note:The obtained exact equivalence is correct.

4.2.3. Utilization of the previous results to define a generic formula to obtain direct

exact equivalences nSNS for polynomial successions nb .

On first previous example we verify that to( ) ( ) ( ) ( ) 221 12 nnQSnnPb nn ====

On second example we see that to( ) ( ) ( ) ( ) nnnQSnnnPb nn ====

332233

By the presented examples and still by others executed, but not presented here, we can

conclude, generically, that if( ) ( ) ( ) ( )nQSnPb mnm

n

1+== .

14


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Note: On two previous resolved examples, the expression of polynomial succession nS does not contain

the constant term 1+ma . This situation is recurrent as we demonstrate next.

Then, considering( ) ( )

11

1

0

1 ... +++ ++++== mm

mmm

n ananananQS and consequently

( ) ( ) ( ) ( ) ( ) 111

0

1

1 1...111 +++

++++== mmmmm

n ananananQS we can write:

( ) ( ) ( ) ++++++++= ++++ 111011101 1...11... mmmmmmmmnn anananaanananaSS ( ) ( ) ( ) ( )1...11...

1

1

01

1

01++++++= ++ nanananananaSS m

mm

m

mm

nn

Then, into( ) ( ) nanananQS

m

mmm

n+++== ++ ...

1

1

0

1the constant term 1+ma has disappeared.

4.2.4. Utilization of the generic formula( ) ( ) ( ) ( )nQSnPb mnm

n

1+== to obtainspecific valid exact equivalences nSNS .

The generic formula( ) ( ) ( ) ( )nQSnPb mnm

n

1+== implies that all the polynomialsuccessions nb respect the TAD and are able to generate valid exact equivalences

nSNS . The practical determination of specific exact equivalences can be executedrecurring to the generic formula and to the Method of Indeterminate Coefficients.

Example: Consider the polynomial succession ( ) ( ) 33 nbnPb nn == The corresponding nS must be ( ) ( )nQSn 4= . Containing all the possible hypotheses for the concrete extensive expression of

( ) ( )nQSn4= and eliminating the last constant term, we can write

( ) ( ) DnCnBnAnnQSn +++==2344

being DCBA ,,, indeterminate coefficients.

The determination of those coefficients can be obtained through the creation andresolution of one system of four equations recurring to the four first equalities of terms

between nS and nNS . So:

To 11 111 ==+++== bNSDCBASn

To 981248162 2122 =+=+=+++== bSSNDCBASn

To 362793927813 3233 =+=+=+++== bSNSDCBASn

To 1006436416642564 4344 =+=+=+++== bSNSDCBASn

Since those four equalities nn NSS = we may write the following system andcorresponding solution. Thus:

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4

2

0

41

21

41

1 0 041 66 42 5 6

3 6392 78 1

9481 6

1

234nnn

S

D

C

B

A

DCBA

DCBA

DCBA

DCBA

n

++=

=

=

=

=

=+++

=+++

=+++

=+++

Then we can write the provisionally exact equivalence:

( )

=

++==1

2343

4

2

n

n

nnnSnNS

Simplified Method of Successive Verifications (to 4>n ) To 2251251002255 5455 =+=+=== bSNSSn (confirms)

To4412162254416

6566

=+=+=== bSNSSn(confirms)

Note: As we have seen, it is possible create a generic method which can be used to calculate valid exact

equivalences nSNS for all the specific polynomial successions nb .In fact, all those successions obey to the TAD and so the adequatedecomposition is also unique.

However, for this functional group it is always SIGSSIGb nn .Then, as the limit of nS is always infinite to all the polynomial successions nb , this situation seems

reduce the interest for the determination of exact equivalences.

However, beside the SIT, there are other reasons of interest for the determination of the nS functional

expression.

In fact, that knowledge will allow calculate at once, with simplicity and precision the numerical value of

the sum of any number of sequent terms of nb . Thus: Possibility of calculation of numerical value of any term of NS with established exact equivalencenSNS

For example, the term of th40 order of oneNSwill be ( )=

=

==40

1

4040

n

n

n SbNS

Then, to the previous exact equivalence ( )

=

++==

1

2343

4

2

n

n

nnnSnNS we have:

6724004

4040240 234

4040 =++

== SNS

Possibility of calculation of the numerical value of the sum of any number of sequential terms ofnb .

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For example, suppose we want calculate the sum 102474645 ... bbbb ++++ of a NSwith establishedvalid exact equivalence nSNS .

Then we can write ( ) ( ) =

=

=

=

==++++102

1

44

1

44102102474645 ...n

n

n

n

nn SSbbbbbb .

Thus, to the same previous exact equivalence it will be:

26613909

4

4444244

4

1021022102...

44102

234234

44102102474645

=

++

++

==++++

SS

SSbbbb

4.3. Another study of attainment of valid exact equivalences nSNS for a secondtype of functional successions. The Rational Successions

4.3.1. Introduction

The rational successions nu are here generically defined by quotients between

polynomial successions or,

( ) ( )( )

( )nR

nP

w

vu

p

m

n

nn == .

The first rational successions with interest for us will be of type:( ) ( )

( ) ( ) ( ) ( )1=

nQnQ

nPu

qq

m

n with ( ) Nqn , , 0Nm and 12 qm

Note: The rational successions whose denominator is the product of two generic sequent terms of the

same polynomial succession may be decomposed according the TAD and, consequently, they are able to

generate valid exact equivalences nSNS .

4.3.2. Generic formularies for valid exact equivalences nSNS for the chosenrational successions.

The attainment of valid exact equivalences nSNS using the nSNSEP , sinceseveral significant rational successions nu with several degrees m and q,judiciously

chosen, originate the following generic formulas to both relationships nn Sb whichare:

Relationships designed as type 1 such as S IC SSS I Sbnn

\

For those types of relationships we obtain the following generic formula:

If

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( )nQ

nRS

nQnQ

nPb

q

q

nqq

m

n ==

1with 221 qmq

Relationships designed as type 2 such as S ISS I SC Sb nn \ orSIGSSIGb

nn

For those types of relationships we obtain the following generic formula:

If

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( )nQ

nRS

nQnQ

nPb

q

qm

nqq

m

n

1

1

+

=

= with pm 2

Note 1: As we may see the presented relationships of type 1 and 2 let outside the relationships of type

SIGSSISbnn = . Forwards, into a special chapter, that study will be made.

17


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Note 2: In a similar way relatively to the valid exact equivalences for polynomial successions, it is

possible demonstrate that the exact equivalences resulting from the rational successions nb , implies that

the nS rational expression has a polynomial numerator without the last constant term.

So, the nS generic expressions can be extensively represented by:

To the relationships of type 1 will beqq

qq

p

qq

nbnbnbnb

nanana

S ++++

+++=

1

1

10

1

1

10

...

...

To relationships of type 2 will beqq

pq

pm

qmqm

nbnbnbnb

nananaS

++++

+++=

+

1

1

10

1

1

0

...

...

4.3.3. Methods for determination of exact equivalences for rational successions.

4.3.3.1. Introduction

The direct relationships nn Sb for determination of specific valid exact equivalences

nSNS may be executed using various methods.We present here the two most important which are:

i) Generic method which consists into recurring to the formulary exposed

behind for the relationships nn Sb of types 1 and 2 and, similarly with polynomial successions, creates and resolves an equation system using

indeterminate coefficients.

ii) TAD method which consists into searching the adequate decomposition of

the rational successions nb . That method is interesting especially to the

relationships nn Sb of type 1.

4.3.3.2. Examples

Example of generic method: Consider( )

( )( ) ( ) ( ) ( ) ( )11

1

2

11

2

+ +== nnnn

nQnQnPbn

The succession S IS Cbn

\ is viable and establishes relationships nn Sb of type 2.Thus, as 2=m and 1=q we will have, according the appropriate formulary.

( ) 111 22

++

=++

=n

BnAnS

nn

nnb nn

The expression of nS has two indeterminate coefficients A and B being necessary a

system of two equation extracted from the successive equalities ( ) nn SNS = . Thus:

To 2

1

21 111 ==

+

== bNSBA

Sn

To3

4

6

5

2

1

3

242 2122 =+=+=

+== bSNS

BASn

The equation system and corresponding solution are:

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19/29

10

1

424

12

+=

=

=

=+

=+

n

n

SB

A

BA

BA

n

We may write the provisionally exact equivalence( )

= +=

++=

1

22

11

1

n

nn

nS

nn

nnNS

Simplified Method of Successive Verifications (to 3>n ) To

4

9

12

11

3

4

4

93 3233 =+=+=== bSNSSn (confirms)

To5

16

20

19

4

9

5

164

4344 =+=+=== bSNSSn (confirms)

Example of TAD method:Consider the succession( ) 22

2

1

13

nn

nnb

n++

=

It is a viable SISbn able to establish relationships nSNS of type 1 with 2=m and 2=q . According the TAD we must search the decomposition of 1= nnn uub or

nnnuub = 1 . We obtain:

( )( )( ) ( )

+

+=

==

++++=

++=

222222

2

1

2313

1

3

1

1

1

13

n

n

n

nbB

A

n

BnA

n

BA n

nn

nnbnn

( ) ( )C

n

nS

n

nu nn ++

+=++=

221

23

1

23

Calculation of C numerical valueAs

( ) ( ) 2

2

211

122

1232

45

43

++=+

++==

==

nnnS

nnSCCubC nn

Thus, provisionally, we may write( ) ( )

= ++

=

++

=1

2

2

22

2

1

2

1

13

n

nn

nnS

nn

nnNS

Note: The rational successions

( ) ( )( ) ( ) ( ) ( )1

=nQnQ

nPb

qq

m

n with 221 qmq , when

decomposable according the TAD, may be decomposed doing( ) ( )

( )

( )( )

( )

( ) ( )( )

( )

( ) ( )( )

( )nQ

nR

nQ

nR

nQnQ

nPq

p

q

p

qq

m

=

1

1

1

with ( ) Nqn , and ( ) 0, Npm .

19


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The sequence of numerical value of Nm placed into interval 221 qmq imposes thesequence of numerical values of 0Np which will be placed into interval 10 qp . Thus:

To 1=q only can be 00 == pm

To 2=q only can be 12 == pm

Example of generic method to the same previous rational successionAs

( )

++=

22

2

1

13

nn

nnbn 2=m and 2=q then the corresponding

( ) 2

2

1++

=n

BnAnS

n

The equation system and the respective solution will be.

( ) 2

2

1

2

1

2

52

3

+

+=

=

=

=+

=+

n

nnS

B

A

BA

BA

n

Then we may write the exact equivalence( ) ( )

= ++

=

++

=1

2

2

22

2

1

2

1

13

n

nn

nnS

nn

nnNS

Note: The results of equivalence nSNS obtained by both former methods are coincident whichbecomes unnecessary its confirmation.

4.3.4. The determination of equivalences nSNS applied to Mengoli Series (MS)The MScan be convergent and, in that case, it is possible determine the respective SIT.

NeitherMShas known equivalences nSNS , today.However, this unknowing can be resolved using the knowledge about the relationships

nn Sb of type 1. Thus, we know that any ( )

=+=

1n

pnn uuMS with Np are

convergent when 0lim = +pnnn uuSCu .

At those circumstances, it will be ( ) ( )=

=+=

pn

n

pnn uuSIT1

lim .

The convergent MS with 1=p , or symbolically, ( )

=+=

1

1

n

nn uuMS are similar

relatively to ( )

= +==

1

1

n

nnnn CuSuuNS by the TAD.

On other hand, it is always possible to transform a MSwith 1>p into a parcelled sum

ofMSwith 1=p as we demonstrate next. Thus:

( ) ( ) +++==

=

= ++++++1 11211 ...n n

pnpnnnnnpnn uuuuuuuuMS

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21/29

( ) ( ) ( ) ( )

=

=

=

=++++++ +++=

1 1 1 1

1211 ...n n n n

pnpnnnnnpnnuuuuuuuu

Then, by the TAD we can write ( )

=++++ +++==

1

21 ...n

pnnnnpnnSSSSuuMS

Example: Consider the =

=

+

+=

1

34

11

1

n

pnn

MS

We may write:

=

=

=

=

+

++

+

++

+

+=

+

+1 1 1 1 41

3

1

3

1

2

1

2

1

1

1

4

1

1

1

n n n n nnnnnnnn

( ) ( ) ( ) ( ) ( )

=

=

=

=

++

+

++

+

++

=

+

+

1 1 1 1 43

1

32

1

21

1

4

1

1

1

n n n n nnnnnnnn

( ) ( ) ( )4433224

1

1

1

1 ++

++

+=

+

+

= n

n

n

n

n

nS

nnnn

II

THE THEORY OF APPROXIMATE EQUIVALENCES nSNS

1. Introduction

The development of certain mathematical realities known for long time but never

explored to the end here proposed allow establish approximate equivalences of typeCuSNS

nn+ in which the signal is reading approximate equal

Those approximate equivalences allow obtain results with great importance to the

general theory ofNS.

The main bases for the proposed new approach are:

Principle of Approximate Equality (PAE)and

Theorem of Approximate Logarithmic Decomposition (TALD)That PAE and TALD jointly with the nSNSPE , and TAD presented behind willconstitute the fundaments for the theory that we analyse next.

1.1. ThePrinciple of Approximate Equality (PAE)

Consider two successions SISun and SISvn such as 1lim =n

n

v

u.

The presented conditions allows assure that to enough high numerical values of n we

can write the approximateequality nn vu .

Example: Consider the successionsn

un

1= and999

1+= nvn

We know that SISun , SISvn and 1lim =n

n

v

u.

Then for pn > we can write the approximate equality nn vu . In fact: To 1111 001.011 vuvun ===

To 1000100010001000 000999.0001.01000 vuvun ===

It is obvious that the effect ofapproximateequality verified already to 1000=n does

accentuate its character to 1000>n and, consequently, the approximate equation

21


22/29

nn vu reaches the absolute equality or nn vu = near from the limit

( ) ( )( )nn vu limlim = .

The evidence of the previous example and the obvious possibility for its generalization

to other pairs of successions nu and nv allow enunciate thePAEas being:

Given two any successions SISbn and SISvn such as 1lim =n

n

v

bthen, to a

certain order pn > it occurs the approximate equality nn vb which is more andmore visible when n , reaching the absolute equality nn vu = on limit

Note 1: Application ofPAEfor approximate equalities of successions.

The possibility of establish approximate equalities between successions nb and nv respecting thePAE

is mainly important when nb define transcendent successions (logartmica, exponential, trigonometric).

Thus, we know that if SIPvn there are, among others, the following notable unitary limits:

( ) ( ) ( ) ( ) ( ) ( ) 1tanlim4;1sinlim3;11lnlim)2(;11lim10000

===+=

n

n

vn

n

vn

n

vn

v

v vv

vv

vv

ve

nnn

n

n

The existence of those known unitary limits can be profited to generate approximate equalities,

respectively as ( ) ( ) ( ) ( ) ( ) ( ) ( ) nnnnnnnv

vvvvvvve n + tan4;sin3;1ln2;11

Examples of approximate equalities:

(1)1

112

ne n ; (2)

ne n

11

1sin

=

; (3)33

111lntan

nn

+ (4)

2

2

4

1

2

1tansin

nn

(5)1

1

1

11ln

1ln

222

2

+

+=

+ nnn

n(6)

n

n

ne n

11

11 +

=+

Note 2: Application ofPAE as complete equality for determination of certain complex limits withtranscendent successions.

Certain limits with transcendent successions very complex may sometimes escape to the normal methods

for resolution. In that case, its analyse looking for the application ofPAEin complete equality may allow

its simplification in order to achieve to a known limit.

Example: Consider the succession

+

+= n

n

n

u

1tan1ln

1

2

21sin1

Determine its limit: We have( ) e

nn

u

n

nn =

+=

+=

2

2

2

1tanln

1lim

2

11lim

1sin1limlim

Note 3: The approximate equality ( ) nn vv +1ln with SIPvn is especially important to thesequence of this article.

1.2. Utilization of thePAEalone or associated with the nSNSEP , or with TADfor determination of approximate equivalences CuSNS nn +

1.2.1. Introduction

The clarification of the nature ofNS with transcendent nb as logartmica, or

exponential or trigonometric successions is many times source of perplexity.

In fact, the normal convergence criteria as comparison, DAlembert or Cauchy not

always are sufficient to clarify the nature of thoseNS.

22


23/29

One known example is the( )

=

=1 ln

1

n nnNS whose divergence only can be

confirmed by the Integral Criteria, extrinsic to the theory ofSN.

The utilization ofPAE alone in some situations or conjugated with nSNSEP , in

other cases or with the TAD in some other cases yet allow clarify not only the nature ofthose types ofNS (avoiding the integral criteria) but also, many times, calculate the

corresponding nS although approximately.

1.2.2. Utilization ofPAEalone to obtaining approximate equivalences.

Consider nb and nv two successions obeying toPAE.

Under that circumstances it seems to be possible write ( ) ( )

=

=

1 1n n

nn vb .

If it is possible establish the exact equivalence ( )

=

+==1n

nnn CuSvNS then we can

write ( )=

+=1n

nnn CuSbNS .

Example: Consider ( )

=

+

=

1

1

1

1n

nneNS . Determine its approximate equivalence.

As by thePAEwe may write ( )

( )1

111

1

++

nne nn we have:

( )

( )

=

=

+

+

11

1

1

1

11

nn

nn

nne . Being

( )

= +=

+

=1 11

1

n

nn

nS

nnNS then:

( )

11

1

1

1

+

=

=

+

nnSeNS n

n

nn

Note: The established approximate equivalence of example does not foresee any constant C. However,

the comparison by difference between the successive terms of NS and corresponding terms of nu

originate a succession of errors ( )n defined by ( ) nnn uSN = .

Thus, the first 6 terms of the succession of errors ( )654321 ,,,,, will be:

To ( ) ( ) 149.05.0649.01 1111111 ==== uNSubNSn

To ( ) 163.0667.083.02 22212 +== ubbNSn

To

( ) ( ) 167.075.0917.0333323 =+==

ubNSNSn

To ( ) ( ) 168.08.0968.04 44434 =+== ubNSNSn

To ( ) ( ) 169.0883.0002.15 55545 +== ubNSNSn

To ( ) ( ) 169.0857.0026.16 66656 +== ubNSNSn

As we see to those first subtractions of terms the succession of errors n increase slowly and seems to

tend for a constant numerical value approximately equal to 169.0 .

Being so we may write ( ) 169.0lim n .

Then instead the initial approximate equivalence( )

=

+

+

=

1

1

1

11

n

n

nm

n

nSeNS (1) will be

more correct and complete write( )

=

+ ++

=

1

1

1

169.01

1n

n

nn

nnSeNS (2)

23


24/29

In fact the expression (2) still being an approximate equivalence but with more quality of approximation

than the expression (1) because the introduction of constant 169.0C into (2).In a general manner the establishment of approximate equivalences by the application ofPAEimply the

appearing of a constant C whose exact numerical value will be given by

( ) ( )[ ]nnn uNSC == limlim .

The determination ofC is, normally, very slow and truly a few interesting. The situation becomes worstwhen NS and corresponding nS are divergent.

Thats why we do not determine the rigorous numerical value of C unless in special cases referringpresumable values when advisable.

1.2.3. Utilization ofPAE in association with nSNSEP , for determination ofapproximate equivalences CuSNS nn + The association betweenPAEand nSNSEP , may establish important approximateequivalences CuSNS nn + .

Example: Consider the succession ( )( )1ln11ln

+

+=

n

nun

Utilization of nSNSEP ,The presented S IC Su

n\ has positive signal to 1>n and ( ) =1lim nu .

Thus, nu is a viable succession. Then we may calculate nb as being:

( )

( )

( )

( )

( ) ( )

( ) ( )

++

=

+

+==

1lnln

ln1ln

ln

1ln

1ln

11ln1

nn

nnb

n

n

n

nbuub

nnnnn

( ) ( )1lnln

11ln

+

+

=nn

nbn

.

Intervention of PAEAccording thePAE,we may write

nn

111ln

+ .

Then

( ) ( ) ( ) ( ) ( ) ( )1lnln1

1lnln

1

1lnln

11ln

+

+

+

+

=nnn

bnn

nbnn

nb nnn

Thus, we may establish the important approximate equivalence

( ) ( )

( )

( )

= ++

+

+= 2 1ln11ln

1lnln

1

nn Cn

n

SnnnNS (1)

Note: Some calculations gives us the conviction that 62.0C . Nevertheless, more important than the knowing of exact value of C is the guarantee that

( ) ( )

=

+

=1 1lnln

1

n nnnNS is convergent.

In fact, until now, the demonstration that the classic( )

=

=2

2ln

1

n nnSN is convergent only could

be made recurring to the Criteria of Integral.

24


25/29

The demonstration that the comparable( ) ( )

=

+

=1 1lnln

1

n nnnNS is convergent can dispense

the Criteria of Integral being necessary only the Criteria of Comparison.

1.2.4. The utilization ofPAE in association with TAD for determination of other

important approximate equivalences CuSNS nn +1.2.4.1. Introduction. The Theorem of Approximate Logarithmic Decomposition

(TALD)

Consider the succession S IC Svn

\= or SIGvn such as ++ =1lim 1n

n

v

v.

If it is so, we may write )1(111

n

nn

n

n

v

vv

v

v += ++ .

Applying neperianos logarithm to equation (1) and according thePAEwe obtain:

( ) ( )n

nn

nn

n

nn

n

n

v

vvvv

v

vv

v

v

+=

++

++ 11

11 lnln1lnln (2)

If we consider into the equation (2) that ( ) nn uv =+1ln , ( ) 1ln = nn uv and

n

nn

nv

vvb

= +1 then the approximate equality (2) is of type 1 nnn uub defining an

approximate decomposition of nb which can be changed into an approximateadequate

decomposition.Thus, according the TAD, we can define for the equation (2) a generic approximate

equivalence nSNS as being ( )

=+

+ +

=

1

1

1 lnn

nn

n

nn CuSu

uuNS

We are now in conditions of enunciate the

Theorem of Approximate Logarithmic Decomposition (TALD)

Given a succession S IC Sun

\ or SIGun such as ++ =

1lim 1

n

n

u

uthen it is

possible establish the Approximate Logarithmic Decomposition

( ) ( )nnn

nn

n uuu

uub lnln 1

1 =

= ++

and, according the TAD, define the approximate

equivalence ( )

=

++ +

=

1

1

1 ln

n

nn

n

nn CuSu

uuNS

2. Utilization ofTALD for determination of approximate equivalences for rational

successions of type

( ) ( )( ) ( )nQ

nPu

p

p

n

1

= which are not studied before and for successions

defined by quotients of non polynomial successions.

2.1. Utilization of TALD for determination of approximate equivalences for

rational successions of type

( ) ( )( ) ( )nQ

nPu

p

p

n

1

=

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26/29

Consider the generic polynomial succession( ) ( )nQu pn = with ( ) Npn , such as

( ) ( )( ) ( )

+=+

11

limnQ

nQp

p

and apply it thePAE.

We obtain

( ) ( ) ( ) ( )

( ) ( )

( ) ( )[ ] ( ) ( )[ ]nQnQnQ

nQnQb

pp

p

pp

n

ln1ln1

++

=.

Then, according the TALD we can write:( ) ( ) ( ) ( )

( ) ( )( ) ( )[ ] CnQS

nQ

nQnQNS pn

np

pp

++

+=

=

1ln1

1

.

Because it is always( ) ( ) ( ) ( ) ( ) ( )nRnQnQ ppp 11 =+ we have:

( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

SISbnQ

nR

nQ

nQnQb np

p

p

pp

n =+

=1

1.

Any

( ) ( )( )

( )

=

=

1

1

n

p

p

nQ

nRNS is always divergent by the Criteria of Comparison.

However, if( ) ( ) ( ) ( ) ( )nQnQnR ppp += 11 , we can determine its approximate

equivalences which give very important results.

First example: The imprecision of EulerConsider the polynomial succession

( ) ( ) nnQun ==1

As++ =

+ 1

1limlim 1

n

n

u

u

n

nwe may write:

( ) ( ) ( ) ( )( ) ( )

( )

=

++

==+

=+

=1

1

11

1ln1111

n

nnnn CnSn

SNn

bn

nnb

nQ

nQnQb


In that case, the exact value ofCshould be ( )[ ]1lnlim1

lim1

+

=

=

nn

Cn

(1).

To avoid execute the direct calculation ofCwhich would be very painful, we can recur

to the known constant ofEuler-Mascheroni given by ( )[ ]nnn

lnlim1

lim1

=

=

(2)

The comparison between both expressions (1) and (2) allow assure that =C .

Then, the complete approximate equivalence will be ( )

=

++

=1

1ln1

n

n nS

n

NS

Note 1: The approximate equality ( )

=+

1

ln1

n

nn

(1) due toEulerin which ( )nun ln= and the

approximate equivalence ( )

=

++

=

1

1ln1

n

n nSn

NS (2) due to the TALD application in

which ( )1ln += nun are not exactly the same because ( ) ( )1lnln + nn .The little discrepancy on both expressions favours the approximate equivalence (2) because the

succession of errors ( )n is smaller than the similar n coming from theEulerexpression.Since the obtained results are enough clarifying we make the comparison only for the three first terms.

Then we have:

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27/29

To

( ) ( ) ( )

( ) ( ) ( )

===

=====

3.02l n1

11l n11

11111111

11111111

ubuN ST A L D

ubuN SE u l e r n

To

( ) ( ) ( )

( ) ( ) ( )

+==

+===

4.03l n2

11

8.02l n2

11

2

22222

22222

uN ST A L D

uN SE u l e r

n

To

( ) ( ) ( )

( ) ( ) ( )

++==

++==

=4 4.04l n

3

1

2

11

7 3.03l n3

1

2

11

3

33333

3333

uN ST A L D

uN SE u l e r

n

Obviously, the sequence of n on both expressions in comparison are tending to ( )577.0 .However, the expression due toEulerproduces greater partial and thus, those

+ .On other hand, the expression due ofTALDproduces smaller partial and thus, those

.

Note 2: The demonstration of the expression due to Eulerneeds of knowledge of Integral Calculus to be

understood.

The expression coming from TALD, demonstrate the same thing but maintaining the necessaryknowledge strictly within the theory ofNS.

Second example: Consider the generic first degree polynomial succession pnun += with Np . Lets search a formula to obtain a generic approximate equivalence to it.

As+

+

++ =+++

= 11

lim1

pn

pn

u

u

pn

pn

we may write( )

pnpn

pnpnbn +

=+

+++=

11.

Then, according TALDwe have ( )

=

+++

+

=1

1ln1

n

pnCpnS

pnNS

Calculation of pC generic expressionThe calculation of pC generic expression is the greater obstacle to establish a genericformula for those equivalences.

However, we can surpass this obstacle recurring to an artifice.

Then,

= = =

=

+

=

+

=1 1 1

11lim

111

n

p

n

p

n nSIT

pnp

nSIT

pnnMS (1)

On other hand we know that:

27


28/29

( ) ( )[ ]

=

=

=+ +++++

+

=

+

1 1 1

1ln1ln1111

n n n

ppnnCpnnSS

pnnpnn

=+ +++

+

+

1 1

1ln

11

n

ppnn Cpn

nSS

pnn (2)

Equalizing the results (1) and (2) in limit we can define the generic expression of PC .

Thus, =

=

=

=

=+

++

+ p

n

pn

n

ppn

Cn

Cpn

n

1 1

11

1

1lnlim

We can write thus the generic formula to those types ofNSas being:

( )

=

=

=

+++

+

=1 1

11ln

1

n

pn

n

nn

pnSpn

NS

Concrete case: Consider the

=

+=

1 4

1

n nNS .

Determine the equivalence.It will be:

( ) ( )

=

=

=

=+ +++ +1

4

1 1

51.15ln4

115ln4

1n

n

n n

nn nSnn

nSn

Third example: Consider the polynomial succession ( ) ( ) 133 +== nnQunAs

( ) ++ =+

++ 1

1

11limlim

3

3

1

n

n

u

u

n

nwe may write:

( ) ( ) ( ) ( )( ) ( )

+

++=

+++

=+

=1

133

1

123313

2

3

323

3

33

n

nnb

n

nnnnb

nQ

nQnQb nnn

( )

=

++++

+

++=

1

23

3

2

233ln1

133

n

nCnnnS

n

nnNS

Calculation of C numerical valueThe calculation ofCwould be painful and with a reduced interest because it can not be

generalized.

However, even without the calculation ofCthe obtained equivalence assures us that this

NSis divergent otherwise like all the othersNSof same type.

2.2. Application ofTALD to studySNwith nb defined by quotients of successions

not exclusively polynomials.

The TALD is also very useful to determine others approximate equivalencesCuSNSnn

+ or, at least, clarify the convergence or divergence of important NSwhose nb are defined by quotients of successions not exclusively polynomials.

First example: Consider the logarithmic succession ( )nun ln= .As

( )

( )++ =

+= 1

ln

1lnlimlim 1

n

n

u

u

n

nwe may write:

( ) ( )

( ) ( ) ( ) ( )nnb

n

nb

n

n

n

bn

nnb nnnn

ln

1

ln

11ln

ln

1ln

ln

ln1ln

+

=

+

=+

=

28


29/29

Applying the TALDwe have( )

( )[ ]

=

++

=1

1lnlnln

1

n

nCnS

nnNS

Determination of C numerical valueThe determination ofCis disinteresting but to any IRC the approximate equivalenceassures the divergence of thisNS.

Note: Until now, the divergence of( )

=

=1 ln

1

n nnSN only could be demonstrated using the

Criteria of Integral. The TALDdemonstrate it dispensing the referred Criteria.

III

FINAL CONCLUSIONS

The article that we finish now, introduces new knowledge about NSwhich may changethe study of this important subject. Thus:

Contradict the generalized idea that only a fewNSmay be summed. Become as redundant the Criteria if Integral to demonstrate the convergence ordivergence of someNSand becomeunnecessary the anticipation of study of the IntegralCalculus to understand the behaviour of anyNS.

Clarify the relationship between the successions andNS. It is a great help to introduce and understand the Integral Calculus. Establish without any obstacle the natural sequence of mathematical matters as beingSuccessions, Numerical Series and Integral Calculus.

subsidies for a generic theory about equivalences

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