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
Calculation of C numerical value
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.
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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+== .
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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.
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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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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 .
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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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( ) ( ) ( ) ( )
=
=
=
=++++++ +++=
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
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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.
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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)
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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.
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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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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
Calculation of C numerical value
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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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:
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( ) ( )[ ]
=
=
=+ +++++
+
=
+
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
+
=
+
=+
=
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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.
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