Hyperreal Structures Arisig From A Ifiite Base Logarithm by Eric Legyel Copyright 996 Abstract This paper presets ew cocepts i the use of ifiite ad ifiitesimal umbers i real aalysis The theory is based upo the hyperreal umber system developed by Abraham Robiso i the 960's i his ivetio of ostadard aalysis The paper begis with a short expositio of the costructio of the hyperreal umber system ad the fudametal results of ostadard aalysis which are used throughout the paper The ew theory which is built upo this foudatio orgaizes the set of hyperreal umbers through structures which deped o a ifiite base logarithm Several ew relatios are itroduced whose properties eable the simplificatio of calculatios ivolvig ifiite ad ifiitesimal umbers The paper explores two areas of applicatio of these results to stadard problems i elemetary calculus The first is to the evaluatio of limits which assume certai idetermiate forms The secod is to the determiatio of covergece of ifiite series Both applicatios provide methods which greatly reduce the amout of computatio ecessary i may situatios
Overview I the 960's, Abraham Robiso developed what is called ostadard aalysis, ad i doig so provided a rigorous foudatio for the use of ifiitesimals i aalysis A ew umber system ow as the set of hyperreal umbers was costructed which icludes the set of real umbers ad also cotais ifiite ad ifiitesimal umbers This paper begis with the costructio of the hyperreals as a set of equivalece classes of sequeces of real umbers Icluded i this itroductory sectio is the method by which relatios defied o the real umbers are exteded to relatios o the hyperreal umbers Through these extesios, it is possible to prove statemets that hold true over the hyperreal umbers which are the ostadard equivalets of statemets that hold true over the real umbers I ostadard aalysis, the proofs of these statemets are usually facilitated by the utilizatio of what is called the trasfer priciple This cocept, however, requires a great deal of developmet of rigorous logic which will ot be eeded i the remaider of the paper Therefore, the trasfer priciple is ot used i the itroductory sectio at all, ad alterate proofs of ostadard results are istead give I Sectio, we begi the study of ew hyperreal structures which orgaize the set of hyperreal umbers ito classes called zoes, ad we itroduce the otios of superiority ad local equality These cocepts deped o the logarithms of the hyperreal umbers tae to a fixed ifiite base We will be usig ifiite ad ifiitesimal umbers i such a way that our calculatios will ot require owledge of how the hyperreal umber system was costructed Nor will our calculatios require us to ow what specific member of the set of hyperreals that we are usig as the ifiite base of our logarithm owig oly that the umber is ifiite will suffice Oce the theory has bee developed i Sectio, we proceed i Sectios 3 ad 4 to preset applicatios i two areas The first applicatio is to the evaluatio of certai limits which assume the idetermiate forms 00,,, ad The methods preseted will provide alteratives to l Hôpital s rule which geerally allow much more efficiet computatio The secod applicatio is to the determiatio of the covergece of ifiite series Two ew covergece tests will be preseted which are aalogous to the compariso test ad limit compariso test As with limit evaluatio, these tests sigificatly reduce the amout of computatio i may situatios
Itroductio to Nostadard Aalysis T his sectio presets the reader with the fudametals of ostadard aalysis which are used throughout the remaider of this paper We begi with the ultrapower costructio of the set of hyperreal umbers ad the proceed to itroduce several relatios ad algebraic structures which are defied o this set This sectio cosists oly of the ecessary bacgroud iformatio which ca be foud i ay itroductory text o ostadard aalysis, most otably [3] May of the proofs icluded i this sectio ca also be foud i the literature, the exceptios beig those which rely o the trasfer priciple Costructio of the Hyperreal Number System The goals of the costructio of the hyperreal umber system are to build a field which cotais a isomorphic copy of the real umbers as a proper subfield ad also cotais ifiite ad ifiitesimal umbers Furthermore, it is desired that the umbers i this ew field obey all of the same laws which hold true over the real umbers A field havig these properties is costructed by usig a free ultrafilter to partitio the set of all sequeces of real umbers ito equivalece classes It is the these equivalece classes which are the elemets of the set of hyperreal umbers Before presetig the actual costructio of the hyperreals, we iclude the defiitio of a filter Defiitio A filter F o a set S is a oempty collectio of subsets of S havig the followig properties (a) F (b) If A F ad B F the A B F (c) If A F ad A B S the B F Note that by (c), a filter F o S always cotais S, ad that by (a) ad (b), o two elemets of F are disjoit 3
A ultrafilter o a ifiite set S is a maximal filter o S The existece of a ultrafilter follows from Zor s Lemma A filter F o S is a ultrafilter if ad oly if it has the followig property (d) If A S the either A F or S A F A ultrafilter U o a ifiite set S is called fixed or pricipal if there exists a S such that U = { A S a A} Ultrafilters which are ot fixed are called free A importat fact is that free ultrafilters caot cotai ay fiite sets By (d), this implies that if U is free the every cofiite subset of S is cotaied i U We ow costruct the set of hyperreal umbers ad prove that it is a liearly ordered field We begi the costructio by choosig a free ultrafilter U o the set of atural umbers N The ultrafilter U is ot explicitly defied sice it does ot matter which free ultrafilter o N that we use The set of all free ultrafilters o N determies a set of isomorphic fields from which we ca choose ay member to be the set of hyperreal umbers Usig U, the hyperreals are costructed by cosiderig the set of all sequeces of real umbers idexed by N ad defiig the followig relatio o this set Defiitio Give two sequeces of real umbers a ad b, a b if ad oly if { N a = b} U The etries of the sequeces a ad b are the said to be equal almost everywhere The phrase almost everywhere (abbreviated ae) is used to sigify that the etries of a sequece have a certai property o some set i the ultrafilter U For istace, if the sequece a has the property that a is a iteger for all i some member of U, the a is said to be a iteger almost everywhere Propositio 3 The relatio is a equivalece relatio Proof Let a, b, ad c be sequeces of real umbers a a, ad thus is reflexive, sice N U Because equality o the reals is symmetric, if a b the b a, ad thus is symmetric For trasitivity, suppose a b ad b c Let A = { N = b} ad B = { N b = c} The A B= { N a = c} Sice A ad B are elemets of the ultrafilter U, A B U Therefore, a c The set of hyperreal umbers, which is deoted by *R, is defied to be the set of all equivalece classes iduced by the equivalece relatio We will use the otatio [ a ] to represet the equivalece class cotaiig a 4
Additio, multiplicatio, ad a orderig are defied o *R as follows Defiitio 4 Let [ a ],[ b ] * R The operatios + (additio) ad (multiplicatio) ad the relatio < (less tha) are defied by (a) [ a ] + [ b ] = [ a + b ] (b) [ a ] [ b ] = [ a b ] (c) [ a ] < [ b ] if ad oly if { N a < b } U Propositio 5 The operatios + ad, ad the relatio < are well-defied Proof Suppose [ a ] = [ b ] ad [ c ] = [ d ], ad let A = { N a = b} ad C = { N c = d} (thus AC, U ) The { N a + c = b + d} = A C U So the two sums are equal elemets of the hyperreals The same argumet holds true for multiplicatio Now suppose [ a ] < [ c ] ad let K = { N a < c} U The the set o which b < c cotais K A ad is therefore i U So [ b ] < [ c ] Let L= { N b < c} The the set o which b < d cotais L C ad is therefore i U Thus [ b ] < [ d ] We ow claim that the set of hyperreal umbers combied with the operatios ad orderig give i defiitio 4 is a liearly ordered field Propositio 6 The structure (* R, +, <, ) is a liearly ordered field Proof I order to show that *R is a field, we eed oly prove that *R cotais iverses sice the associative, commutative, ad distributive laws as well as closure are iherited from the real umbers Let [ a ] * R such that [ a ] [ 0,0,0, ] The the set { N a = 0} U Thus we ow that the complemet of this set is i U So defie b by 0, if a = 0; b = a, if a 0 The the product [ a ][ b ] is equivalet to sice { N ab = } = { N a 0} U Therefore a is ivertible ad *R is a field Now let [ a ],[ b ] * R such that [ a ] [ b ] To prove that *R is liearly ordered, we must show that either [ a ] < [ b ] or [ b ] < [ a ] Let A = { N a < b }, B = { N b < a }, ad E = { N a = b } Sice 5
[ a ] [ b ], E U So the complemet of E, which is A B, is i U If A U, the [ a ] < [ b ] If A U, the B E U, i which case ( A B) ( B E) = B U, so [ b ] < [ a ] Thus [ a ] ad [ b ] are ordered Now that we have show that *R is a liearly ordered field, we wish to show that *R has a proper subfield which is isomorphic to R We embed the reals i the hyperreals by defiig the map θ : R * R by θ ( r) = [ rrr,,, ] To show that θ maps R to a proper subfield of *R, ote that [,,3, ] is ot equivalet to the image of ay real umber This equivalece class is actually a example of a ifiite umber as is defied below Ifiite ad Ifiitesimal Numbers Whether a umber is ifiite or ifiitesimal is idepedet of the umber's sig, so the defiitios of ifiite ad ifiitesimal umbers will ivolve absolute values Absolute value is a fuctio that is already defied o the real umbers which we eed to exted to the hyperreal umbers To do this, we simply apply the absolute value etrywise to a equivalece class represetative i *R Much more will be said about extedig fuctios from the real umbers to the hyperreal umbers shortly, but right ow we oly eed the followig defiitio Defiitio 7 For all [ a ] * R, [ a ] = [ a ] Note that this defiitio is equivalet to the hyperreal aalog of the defiitio of absolute value o the reals, [ a ], if [ a ] 0; [ a ] = [ a ], if [ a ] < 0 This is because if [ a ] 0 the a = a o the same set i the ultrafilter for which a 0, ad if [ a ] < 0 the a = a o the same set i the ultrafilter for which a < 0 From this poit o, we will use sigle letters to deote elemets of *R Whe we spea of a real umber r * R, we mea the equivalece class [ rrr,,, ] Although techically speaig, the sets R, Q, Z, ad N are ot true subsets of the 6
hyperreals, we will use these symbols to refer to the images of these sets embedded i *R We ca ow defie what is meas for a umber to be ifiite or ifiitesimal Defiitio 8 Let a * R The a is ifiite if ad oly if a > r for every positive real umber r If a is ot ifiite the it is fiite We call a ifiitesimal if ad oly if a < r for every positive real umber r Note that the set of fiite umbers is a subrig of *R ad the set of ifiitesimals is a ideal of the fiite umbers More importatly, ote that the set of hyperreal umbers is oarchimedea Not every bouded subset of *R is guarateed to have a least upper boud or greatest lower boud For example, the set of ifiite umbers is bouded below by ay fiite umber but has o greatest lower boud We ow itroduce a equivalece relatio which associates umbers which oly differ by a ifiitesimal Defiitio 9 Give xy, * R, x ad y are ear or ifiitely close if ad oly if x y is a ifiitesimal I this case, we write x y The equivalece classes iduced by are called moads Thus, the moad about a umber a * R, writte m( a ) is defied by m( a) = { x * R x a} *R is also partitioed ito classes whose elemets differ by fiite amouts Defiitio 0 The galaxy about a umber a * R, writte G( a ), is defied by G( a) = { x * R x a is fiite} Usig moads ad galaxies, we ca use m ( 0) to represet the set of all ifiitesimal umbers ad G ( 0) to represet the set of all fiite umbers As show below, the set of real umbers is isomorphic to the quotiet rig G( 0) m ( 0) Numbers i *R which are ot images of real umbers are called ostadard Nostadard fiite umbers are always ifiitely close to exactly oe real umber, which is called its stadard part Propositio Let a be a fiite hyperreal umber The there exists a uique real umber r such that r a Proof Let A = { x R x a} Sice a is fiite, A is oempty ad is bouded above Let r be the least upper boud of A For ay real ε > 0, r ε A ad r+ ε A ad thus r ε a< r+ ε So r a ε from which is follows that r a To show that this r is uique, suppose that there exists a 7
real umber s such that s a The sice is trasitive, s r So s r < ε for every real ε > 0, ad thus s = r Defiitio Let a G( 0) The the stadard part of a, deoted by st ( a ) or a, is the real umber r such that a r The fuctio st is called the stadard part map The stadard part map is easily show to be a order preservig homomorphism from G ( 0) oto R with erel m ( 0) The quatity x st ( x) is sometimes called the ostadard part of x 3 Relatios ad *-Trasforms We ow discuss the method through which fuctios defied o the set of real umbers are exteded to the hyperreals The method actually applies to ay arbitrary relatio defied o R, the set of which icludes all of the fuctios defied o R The process of extedig a relatio from R to *R is called a *-trasform for which the geeral defiitio is ext give Defiitio 3 Let P be a -ary relatio o R The the *-trasform of P, deoted by *P is the set of all -tuples ([ ( a) ],[ ( a) ],,[ ( a i i ) i ]) ( * R ) satisfyig { i N (( a),( a),,( a ) ) P} U i i i A -ary relatio o R is simply a subset of R Thus, uary relatios o R are just subsets of R Equality, the orderig give by <, ad fuctios of oe variable are examples of biary relatios I order to acquire a more ituitive feelig for how relatios o R are exteded to *R, we cosider a few examples Let A R The A is a uary relatio o R, so *A cosists of those elemets [ a ] * R such that { N a A} U (ie, a A almost everywhere) It should ow be clear why the otatio *R is used to represet the set of hyperreal umbers, for if A = R the { N a A} = N U, so *R cosists of all equivalece classes represeted by ay sequece of real umbers The set *N is called the set of hyperatural umbers ad cosists of the umbers [ a ] * R for which a N almost everywhere Liewise, * Z= {[ a ] * R a Z ae } ad * Q= {[ a ] * R a Q ae } These are called the hyperitegers ad hyperratioals respectively Propositio 4 Let S R be a set which cotais a ifiite subset of N The *S cotais ifiite umbers 8
Proof Let A be a ifiite subset of N cotaied i S We costruct a ifiite umber i *S as follows Let a be the least elemet of A ad the choose each a to be the least elemet of the set A { a, a,, a } Sice A has o upper boud, the sequece a must have the property that give ay positive real umber r, there exists a N N such that for all > N, we have a > r Thus a > r almost everywhere, satisfyig the requiremet for [ a ] * S to be ifiite Give a set S * R, we use the otatio S to represet the set of ifiite umbers i S Thus, * Z is the set of ifiite hyperitegers i *R Let us cosider equality o R for a momet as a set of duplets ad write ( ab, ) E if ad oly if a = b The *E is the set of all duplets ([ a ],[ ]) (* ) b R satisfyig { N a = b} U, which is exactly how we defied equality o the hyperreals A fuctio f o R of variables ca be thought of as a set of ( + ) -tuples havig the property that if ( c, c,, c, a) f ad ( c, c,, c, b) f, the a = b The *-trasform of f is the set of all ( + ) -tuples of the form + ( c ), c,, c, a * R satisfyig ([ ] [ ( ) ] [ ( ) ] [ ]) ( ) i i i { i N (( c ),( c ),,( c ), a ) f} U i i i So if f is a fuctio of oe variable, the f ([ c ]) [ f ( c ) ] * i = i That is, we just let f operate etrywise o a equivalece class represetative of a umber i *R Note that operatios such as additio ad multiplicatio are actually fuctios of two variables I the ext sectio, we will be taig logarithms of hyperreal umbers usig a hyperreal umber for the base The logarithm is also a fuctio of two variables so we have *log[ b ][ a ] log = b a If we choose [ b ] = [,,3, ] ad [ a ] = [,,, ], the the first etry of *log [ b ] [ a ] is log, which is udefied This is acceptable, however, sice we ca assig ay value we wish to the first etry ad the fuctio will still hold true o a set i the ultrafilter U As log as the *-trasform of a fuctio is defied almost everywhere for the etries of a equivalece class represetative a, it is defied for [ a ] i 9
4 Sequeces ad Series Sequeces ad series of hyperreal umbers will be a importat area of study later i this paper Whe we spea of a sequece of hyperreal umbers, we are actually talig about a sequece of sequeces of real umbers, ad this sequece is idexed ot by the atural umbers, but by the hyperatural umbers Below, we examie how sequeces of real umbers idexed by the atural umbers are *-trasformed to sequeces of hyperreal umbers idexed by the hyperatural umbers Let s N be a sequece of real umbers This sequece is actually a fuctio s : N R where s( ) = s, so we ca thi of s as the set of duplets {(, s), (, s), ( 3, s 3), } The *-trasform of s is the fuctio * s :* N * R where * s( [ a ]) = [ s( a) ] for ay [ a ] * N The sequece * s * N is a extesio of the sequece s N i that for ay N, * s is simply the image of s i the hyperreals If a sequece s teds to a limit i the real umbers, the the sequece * s teds to the same limit i the hyperreal umbers This is prove shortly, but first we eed to discuss a little otatio The symbol is used i the real umber system to deote that which is potetially arbitrarily large Thus the expressio lim s deotes the value that s approaches as becomes a arbitrarily large atural umber I the hyperreal umber system, the symbol has the same meaig, but by arbitrarily large, we mea eve larger tha ay ifiite umber i *R So the expressio lim * s deotes the value that * s approaches as becomes a arbitrarily large hyperatural umber We ow have the followig propositio Propositio 5 Suppose that lim s = L for some real umber L The lim * s = L Proof The fact that lim s = L meas that give ε > 0 there exists a N N such that for all > N, s L < ε Now let ε be a positive hyperreal umber ad let ε i be a equivalece class represetative of ε We defie the hyperatural umber N = [ N i ] by choosig N i to be ay atural umber for which s L < ε i for all > Ni The for ay hyperatural umber > N, * s L < ε So lim * s = L More importat to this paper are ifiite series of hyperreal umbers We defie a ifiite series i terms of its sequece of partial sums Cosider the stadard sequece 0
s( x,, x,, xm) = ax (,, x,, xm), = which is defied as a fuctio of the atural idex ad real umbers x, x,, xm The sum of the ifiite series is represeted by lim s( x,, x,, xm) The *-trasform of s is the sequece * s( x,, x,, xm) = * * ax (,, x,, xm), = where x, x,, xm are hyperreal umbers ad the right had side represets the defiitio of the ostadard summatio from to a hyperatural umber This summatio is writte i equivalece class form as i * * a(, x, x,, xm) = a(, ( x),( x),,( xm) ) i i i = = A importat additio to this defiitio is the requiremet that for each of the hyperreal umbers x, x,, xm, oly oe equivalece class represetative may be used throughout the etire summatio That is, to calculate s( x,, x,, xm ) for a sigle hyperatural umber, we first choose equivalece class represetatives for each of the x, x,, xm ad use the same represetatives each time that we evaluate * a(, x, x,, xm ) i the summatio Without this restrictio, is it possible to fid ill-defied summatios whe is ifiite by choosig differet equivalece class represetatives for each idex The sum of the ifiite series * * a(, x, x,, xm ) = is defied to be lim * s( x,, x,, xm) where approaches ifiity through the hyperatural umbers If this limit exists ad is equal to ay hyperreal umber, icludig ifiite umbers, the the series is said to be coverget We cosider a example that is used later i this paper Let s( abt,,, ) represet the sequece of partial sums of the biomial series for ( a+ b) t This ca be writte as the summatio t t s( abt,,, ) = a b =
I order to use the biomial expasio i the case where a, b, ad t are hyperreal umbers, we eed the *-trasform of s, which is give by t t * s(, a, b, t) = * a b = i ti ti = ai bi = I this summatio, we must use fixed pre-chose equivalece class represetatives for a, b, ad t for every idex 5 Limits Oe fial result of ostadard aalysis that is eeded i this paper is the followig statemet about the limit of a fuctio at a poit or at ifiity Propositio 6 Let L be a real umber (a) lim x a f ( x) = L if ad oly if * f ( x) L for all x m( a) { a} (b) lim x f ( x) = Lif ad oly if * f ( x) L for all positive ifiite x Proof All parts are prove by cotrapositive (a) Suppose * f ( z) / L for some z a We wat to show that there exists a ε > 0 such that for ay δ > 0 there exists a x such that x a < δ, but f ( x) L ε Sice * f ( z) / L, we have * f ( z) L 0 If * f ( z ) is fiite, choose ε = st ( * f ( z) L), ad if * f ( z ) is ifiite, choose ε to be ay positive real umber Let z be a equivalece class represetative of z Sice z a, we ow that We also ow that Z = { N z a < δ } U F = { N f ( z ) L ε} U Choose N Z F ad let x = zn The x a < δ, but f ( x) L ε Therefore lim x a f ( x) L
Now we assume lim x a f ( x) L The there exists a ε > 0 such that for every δ > 0 we ca fid x such that x a < δ, but f ( x) L ε We costruct a sequece x by choosig each x such that x a < ad f ( x ) L ε The we have [ x ] a sice for ay positive real umber r, x a < r for all > N where N is the least atural umber satisfyig N < r But * f ([ x ]) / L because f ( x ) L ε for all N So we have foud a umber ifiitely close to a for which the fuctio value is ot ifiitely close to L (b) Suppose * f ( z) / L for some positive ifiite umber z We wat to show that there exists a ε > 0 such that for ay m N there exists a x > m such that f ( x) L ε As i part (a), * f ( z) L 0 So if * f ( z ) is fiite, choose ε = st ( * f ( z) L), ad if * f ( z ) is ifiite, choose ε to be ay positive real umber Let z be a equivalece class represetative of z Sice z is ifiite, we ow that We also ow that Z = { N z > m} U F = { N f ( z ) L ε} U Choose N Z F ad let x = zn The x > m, but f ( x) L ε Therefore lim x f ( x) L Now we assume lim x f ( x) L The there exists a ε > 0 such that for every N N we ca fid x > N such that f ( x) L ε We costruct a sequece x by choosig each x such that x > ad f ( x ) L ε The [ x ] is ifiite sice for ay positive real umber r, x > r for all > r But * f ([ x ]) / L because f ( x ) L ε for all N So we have foud a ifiite umber for which the fuctio value is ot ifiitely close to L 3
Developmet of New Hyperreal Structures T his sectio itroduces the cocept of superiority ad the equivalece relatio called local equality May structures which arise i the set of hyperreal umbers due to these cocepts are examied alog with their relatioships to existig structures Succeedig sectios discuss applicatios of the properties of these ew structures The followig otatio is used i this ad later sectios The set of all ifiite umbers i *R will be deoted by T, ad the set of all ifiitesimal umbers i *R will be deoted by S The symbols ad will be used to mea less tha or ifiitely close to ad greater tha or ifiitely close to respectively The symbols ad will be used to mea less tha but ot ifiitely close to ad greater tha but ot ifiitely close to respectively Orders of Numbers ad the Superiority Relatio The order of a umber ad all subsequetly depedet relatios are based upo the choice of a positive ifiite umber ω We do ot give a explicit equivalece class represetative for ω sice all calculatios that we perform which ivolve ω do ot actually rely o its value Thus all of the relatios ad structures defied i this sectio have the same properties for every possible choice of ω We eed oly remember that durig ay oe calculatio that ω represets a costat positive ifiite hyperreal umber This method allows us to perform computatios i the hyperreal umbers ad at the same time abstract beyod the eed to ow aythig about the ultrafilter used to costruct the hyperreals or the sequeces that represet equivalece classes All symbolic maipulatio is doe with real umbers ad fuctios of ω We use the letter α to represet ω for ay choice of ω The arbitrary ature of ω ad α allows us to show i a sigle calculatio that certai properties hold for all positive ifiite or ifiitesimal umbers We first defie a equivalece relatio that partitios *R ito what are called zoes This is doe by cosiderig the logarithm base ω of each hyperreal umber We will use the symbol log to represet the fuctio *log o *R as well as the fuctio log o R 4
Defiitio The order of a umber a * R, writte ord( a ), is defied as ord( a) = log a The order of 0 is defied to be ω The order of a umber represets the size of the umber o a scale which trasceds ifiitesimal, fiite, ad ifiite umbers Although this order depeds upo the choice of ω, we have the followig property that does ot deped o the value of ω, but oly o the fact that ω is ifiite Lemma All umbers i G( 0) m( 0) (ie, all fiite o-ifiitesimals) have ifiitesimal order Proof Let x G( 0) m( 0) Sice x is ot ifiitesimal or ifiite, there exist positive real umbers a ad b such that a< x < b For every positive real umber r, ord( r) = l r lω S Sice the logarithm is a strictly icreasig fuctio, we have ord( a) < ord( x) < ord( b) So x has ifiitesimal order Clearly, every ifiite umber has positive order ad every ifiitesimal umber has egative order What is iterestig is that although most ifiite ad ifiitesimal umbers do ot have ifiitesimal order, there are elemets of T ad S that do, ad these umbers are give the followig special ames Defiitio 3 If t T ad ord( t) S, the t is called semi-ifiite The set of ifiite umbers less the set of semi-ifiite umbers is deoted by T Defiitio 4 If s S ad ord( s) S, the s is called semi-ifiitesimal The set of ifiitesimal umbers less the set of semi-ifiitesimal umbers is deoted by S A example of a semi-ifiite umber is lω whose order is l lω lω All semi-ifiitesimal umbers are reciprocals of semi-ifiite umbers ad viceversa, so lω is semi-ifiitesimal Properties of semi-ifiite ad semiifiitesimal umbers are idetified throughout this sectio The order of each hyperreal umber is used to defie the followig relatio o *R Defiitio 5 Give ab, * R, we say a ad b are isometric ad write a = b if ad oly if ord( a) ord( b) The relatio = is a equivalece relatio sice S is a group uder additio The ame isometric is used because two umbers whose orders differ oly by a 5
ifiitesimal are of the same geeral size whe cosidered as members of the vast ifiitesimal ad ifiite extet of the set of hyperreal umbers We ow itroduce the superiority relatio that is used to relate two umbers which are ot isometric Defiitio 6 Let ab, * R such that a=/ b If ord( a) ord( b), the we say a is iferior to b ad write a< b, ad if ord( a) ord( b), the we say a is superior to b ad write a> b The symbol is used to mea iferior to or isometric to, ad the symbol is used to mea superior to or isometric to Note that 0 < a for all ozero a * R Also ote that if a b the a b This property will be useful i several proofs later i this paper If two umbers a ad b i *R are related by =, <, or >, the multiplicatio of both by ay ozero x * R preserves this relatio Lemma 7 Let abx,, * R The a = b implies ax = bx, ad if x 0 the a< b implies ax < bx Proof Suppose a = b The ord( a) ord( b) Addig ord( x ) to both sides, we have ord( a) + ord( x) ord( b) + ord( x) Therefore ord( ax) ord( bx ), so ax = bx Now suppose a< b The ord( a) ord( b), so by a similar argumet ord( ax) ord( bx) ad thus ax < bx Additio of equals does ot preserve the relatios =, <, ad > Cosider as a example that + α = + α, but if we add to both sides the we would have a= a, which is ot true Zoes ad Worlds A equivalece class created by the equivalece relatio = is give the followig ame Defiitio 8 The equivalece class iduced by = cotaiig a * R is called the zoe about a This is writte zoe ( a) = { x * R x= a} We defie zoe( 0) = { 0} The zoe about a is the set of all umbers whose order is i the moad about ord( a ) Thus it is possible to express the zoe about a as 6
( ord( )) ( ord( )) zoe ( a) = ω m a ω m a Note that sice ord( x) = ord( x) for all x * R, if x zoe( a) the x zoe( a) Also ote that sice ord( ) = 0, zoe( ) is the set of all elemets of *R havig ifiitesimal order Therefore zoe( ) cotais all of the ozero real umbers Theorem 9 zoe( ) is a multiplicative subgroup of *R Proof We eed to prove closure ad cotaimet of iverses Let ab, zoe( ) The ord( a) S ad ord( b) S Therefore, ord( ab ) = ord( a) + ord( b) S So ab zoe() ad thus zoe( ) is closed uder multiplicatio zoe( ) cotais multiplicative iverses sice for all x * R, ord x = ord x ad S cotais additive iverses ( ) ( ) Every zoe is a coset of zoe( ) We therefore have the properties that for all a * R, zoe( a) = azoe( ) ad zoe( a ) is closed uder multiplicatio by elemets of zoe( ) All semi-ifiite ad semi-ifiitesimal umbers are cotaied i zoe( ) We ca therefore express the set of semi-ifiite umbers as T zoe( ), ad we ca express the set of semi-ifiitesimal umbers as S zoe( ) No zoe (excludig zoe( 0 )) is closed uder additio sice for ay b< a, a+ b zoe( a), but a+ b a zoe( a) The smallest additive subgroup of *R cotaiig zoe( a ) therefore cotais every umber which is iferior to a This set is give the followig ame Defiitio 0 The world about a * R, deoted W ( a ), is defied by W( a) = { x * R x a} We use the otatio W0 ( a ) to represet the set { x * R x< a} W0 ( a ) ca be thought of as the iterior of the set of zoes cotaied i W( a ), ad zoe( a ) ca be thought of as the boudary of the set of zoes cotaied i W( a ) We thus have for the zoe about a the alterate otatio W ( a) Theorem W( a ) ad W ( a ) are groups uder additio for all a * R 0 Proof 0 W( a) ad 0 W0 ( a) because 0 a for all a * R Each zoe which is cotaied i W( a ) cotais its ow additive iverses We are left with provig closure, which we first prove for the W( a ) case 7
Let x, y W( a) Without loss of geerality, we ca assume that x y We will show that x + y W( x) W( a) This meas we must show that ord( x + y) ord( x) Let z = max ( x, y ) The ord( x+ y) = logω x+ y logω ( x + y ) logω z = logω + logω z ord ( z) () If x = y the z = x If x > y the z = x I both cases, ord( z) ord( x) Therefore, from the above equatio, ord( x + y) ord( x) Closure of W0 ( a ) is prove i the exact same maer sice for x, y W ( a) with x y, W( x) W ( a) 0 0 A importat observatio is that if x > y the x + y= x sice otherwise, if x + y = z< x the x = z y< x, a cotradictio This is the fudametal cocept behid what we soo defie to be local equality Theorem W( a ) ad W ( a ) are rigs if ad oly if a 0 Proof Assume a By Theorem, W( a ) ad W0 ( a ) are additive subgroups of *R We eed oly prove closure uder multiplicatio Let x, y W( a) Sice a ad both x a ad y a, ord( x) 0 ad ord( y) 0 Therefore, ord( xy) = ord( x) + ord ( y) 0 So xy W( a) ad W( a ) is thus closed uder multiplicatio For x, y W0 ( a), we have the simpler situatio that ord( x ) < 0 ad ord( y ) < 0 Therefore, ord( xy ) < 0 ad we have xy W0 ( a) Now suppose that a > The ord( a) So we have ord( a ) = ord( a) ord( a) ad thus a W( a) Therefore W( a ) is ot closed uder multiplicatio Also, W0 ( a ) is ot closed sice if we let b = a W0 ( a), the b W ( a) 0 It turs out that for ay two rigs of the type metioed i Theorem, the smaller subrig is a ideal of the larger Theorem 3 Let ab, * R such that a< b The (a) W( a) W ( b) 8
(b) W ( a) W( b) 0 (c) W( a) W ( b) (d) W ( a) W ( b) 0 0 0 Proof (a) Let x W( a) ad y W( b) The ord( x) 0 (sice a < ) ad ord( y) 0 So ord( xy) = ord( x) + ord( y) ord( x), which implies xy x, ad thus xy W ( a) (b) Idetical to (a) (c) Similar to (a), except ord( y) 0 So ord( xy) ord( x), which implies xy< x (d) Idetical to (c) Theorem 4 W ( ) is a maximal ideal of W ( ) 0 Proof Suppose there exists a ideal I of W ( ) with W ( 0 ) I W ( ) The there exists x I such that x = ad is therefore a elemet of zoe( ) But by Theorem 9, zoe( ) cotais multiplicative iverses, so x is a uit ad thus I = W( ) So W ( ) must be maximal 0 We have ow costructed a field W( ) W 0 ( ) cosiderably smaller tha *R that still cotais a isomorphic copy of the real umbers through the map r [ rrr,,, ] + W0 ( ) ad also cotais ifiite ad ifiitesimal umbers such as lω + W ( ) ad lω + W ( ) 0 0 3 Local Equality The cocept of local equality ivolves choosig a level to wor o ad equatig ay two umbers i *R that differ by a isigificat amout with respect to this level For example, o the fiite level (ie, the level of ), ay two umbers that differ by a sufficietly small ifiitesimal are cosidered to be locally equal The precise defiitio is as follows Defiitio 5 Let ab,, * R with 0 We say a is locally equal to b o the level of ad write a= b if ad oly if a b< 9
Local equality o the level of is a equivalece relatio sice 0 < for all * R ad W0 ( ) is closed uder additio by Theorem Oe immediate observatio is that if x < the x = 0 If a= b, the obviously a+ c= b+ c for ay c * R Also, it follows immediately from Lemma 7 that if a= b the c ac = bc If two umbers are locally equal o the level of the they are also locally equal o the level of ay elemet of zoe( ) Replacig the level represetative with a isometric umber does ot chage aythig Local equality is also preserved if the level represetative is replaced by a superior umber I geeral, if a= b ad m m, the a= b a Lemma 6 If a= b the a = b a Proof If a= b the a b< a So we ca write b= a+ ε where ε < a Sice a+ ε = a, we ow b= a a b A immediate cosequece of this lemma is that if a= b the a= b This a a also lets us mae the statemet that if a= b the a= b This is because if a we divide both sides of a= b by a, we have = ba The dividig both sides b a by b gives us b= a By the lemma, this is equivalet to a= b This property of local equality is useful i Sectio 4 Here are some example local equalities α = α ω 3= 4 + α = + lω 5 6 5 7 7 Note that i the last example give above, eve though lω is a ifiitesimal, it is too large to be igored o the fiite level The reaso is that lω is a semiifiitesimal ad semi-ifiitesimals are ot iferior to ay fiite umber The equivalece classes of local equality o the level of are aalogous to moads Numbers that are i m( a ) but ot i { x x = a} are exactly those which differ from a by a semi-ifiitesimal So x= y implies x y, but the coverse is ot ecessarily true 0
4 Summary All of the structures ad relatios itroduced i this sectio deped upo a positive ifiite umber ω whose exact value is left udefied but is cosidered to be held costat throughout ay statemet, proof, calculatio, etc Wheever the umber ω appears i a expressio, it is assumed to be the same ω that ay relatios appearig i the expressio such as iferiority ad local equality deped o through the order fuctio The arbitrary ature of ω allows us to coclude statemets such as if f ( ω ) L idepedetly of the choice of ω, the f ( t) L for all positive ifiite umbers t This becomes a ivaluable tool whe used i cojuctio with local equality for evaluatig limits which assume a idetermiate form ad also for testig ifiite series for covergece These topics are discussed i the ext two sectios
3 Evaluatio of Limits We ow use the cocept of local equality to develop a system for evaluatig certai types of limits at poits where they assume idetermiate forms The methods preseted are alteratives to l Hôpital s Rule which are geerally easier to use ad i most cases allow much faster computatio Several examples are give which demostrate the ew methods o limits which assume the idetermiate forms 00,,, ad 3 Prelimiary Theory Nostadard aalysis tells us that lim x f ( x) exists ad is equal to L if * f ( x) L for all positive ifiite x * R (see Sectio 5) Whe evaluatig limits of this form, we will be able to show that * f ( ω ) = L idepedetly of the choice of ω Thus, i a sigle calculatio, we will be able to show that * f ( x) L for every ifiite umber ω This implies that lim x f ( x) = L A similar method will be used for limits of the form lim x 0 f ( x) I this case, we eed to show that * f ( x) L for all ifiitesimal x We will be able to obtai * f ( α ) = L idepedetly of the choice of α, from which is follows that lim x 0 f ( x) = L The way i which we will use local equality at first is by usig the fact that give p( x) R [ x], if we choose a S T, the every term of p( x ) evaluated at a belogs to a differet zoe Thus we have properties such as p( ω ) is locally equal o its ow level to the highest degree term of p( x ) evaluated at ω The followig lemma becomes useful i several places where we are worig with ifiite series of ifiitesimals (The * is omitted from the summatios i this sectio all summatios are ostadard) Lemma 3 If β is a ifiitesimal, the β = β = Proof We have the equatio
This tells us that β β = β = β β = = = β < β = = β = β Sice β itself is a summad, β = β = (3) I may of the examples i this sectio, we use power series to represet expoetial ad trigoometric fuctios These series are the evaluated at α, maig it possible to use the followig lemma Lemma 3 Suppose β < ad let a N be a sequece of real umbers such that a is bouded by some real umber m The for ay fiite, β * a * β = a β = = Proof We eed to show that a β < β We ca write this sum as By Lemma 3, This tells us that * = + * aβ * aβ = + = + = β = + = β * a+ = β = β = m β Therefore, = * a β + = β β (3) m β β = β (33) 3
+ * aβ β = + (34) Sice β <, β + < β, so the etire sum is iferior to β 3 α 3 This lemma allows us to mae statemets such as siα = α α 6 3 Limits of the Form 0 0 ad For limits which assume the form 00 or, we eed to examie the properties of local equality as it pertais to quotiets ab If a< b, the ab= 0, ad if a> b, the ab is ifiite For the remaiig case, a = b, we have the followig theorem Theorem 33 Let a, a, b, ad b be ozero elemets of *R Suppose we a b a have the local equalities a = a ad b = b The a b a = b b Proof Write a = a + ε a where ε a < a ad write b = b + ε b where ε b < b Multiplyig ε a by b ad ε b by a, we have the relatios b ε a < ab ad aε b < ab Sice W0( ab ) is closed uder additio, a ε b b ε a < ab By Lemma 6, b = b So we ca write a b < ab (35) ε b ε a The left side of this equatio ca be rewritte by addig ad subtractig ab to obtai a ( b + ε ) b ( a + ε ) < ab (36) b a Replacig b + ε b with b ad a + ε a with a ad the dividig both sides by bb (usig Lemma 7), we have ab ab a < (37) bb b The left side of this equatio is simply b b, so the proof is complete sice a this is ow the defiitio of a b a = b b A trivial example of the applicatio of Theorem 33 is a quotiet of polyomials If we have lim x p( x) q( x), the we substitute ω for x i both p( x ) ad a a 4
q( x ) If p( ω ) = q( ω ), which is true if ad oly if p ad q have the same degree, the the limit is equal to the quotiet of the leadig coefficiets of p ad q Followig are two slightly less trivial examples i which trigoometric fuctios appear si x Example 34 lim x x + x Solutio We replace the sie fuctio with its power series ad evaluate at α to obtai α α + + 3 5 α 3! 5! α + α 3 5 α α α By Lemma 3, α + + = α ad 3! 5! Theorem 33 as follows α 3 5 α α 3! + 5! + α = = α + α α α α + α = α, so we ca apply si x Sice this local equality holds for ay choice of α, we ow lim 0 + = A similar example ivolvig the cosie fuctio is give ext cosx Example 35 lim x 0 4 x 3x x x x Solutio As above, we replace the cosie fuctio with its power series ad evaluate at α to obtai ( ) α α 4 4! + 4! + α α! 4! + = 4 4 α 3α α 3α This time we have Theorem 33, 4 α + =! 4! 4 α ad α 3α = α, so applyig α α α + = = α 3α α 4! 4! 4 α α α cosx Agai, this local equality is idepedet of α So lim 4 = x 0 x 3x 5
Evaluatig the limit i Example 35 would require two applicatios of l Hôpital s rule However, usig local equality eables the limit to be evaluated i a sigle step oce the trigoometric fuctio is replaced by its power series 33 Limits of the Form Far more iterestig situatios arise whe we examie limits which assume the form For this case, we eed the followig theorem Theorem 36 Let a * R ad suppose a = t for some positive ifiite t t t a t umber t * R The for ay b * R satisfyig b= a, we have a = b Proof Let ε = b a ad assume ε 0 Expadig ( a + ε ) t with the biomial theorem, we obtai t t t t b = ( a+ ε ) = a ε (38) = 0 For = 0, the summad is simply a t We wish to show that the sum of the t terms for is iferior to a So we rewrite equatio (38) as t t t t b = a + a ε (39) = ad show that this sum for satisfies the hypothesis of Lemma 3 (We will actually be applyig this lemma twice) t The biomial coefficiet ( ) = t( t )( t ) ( t + )! is a degree polyomial i t Whe the umerator is expressed as a sum of powers of t, the coefficiet of the degree j term is give by the Stirlig umber of the first id s(, j ) where s(, j ) = 0 if j < or j > Thus we ca write equatio (39) as t t t s(, j) j b = a + a ε t = j=! (30) Sice ε < t, we ca write ε = β t where β < Replacig ε with β t ad factorig t out of the ier sum, we have 6
t t t s(, j) j b = a + a β t = j=! (3) We ow reverse the order of summatio for the ier sum through the map j + j so that the terms are arraged i the order ecessary for the applicatio of Lemma 3 This gives us For coveiece, we defie t t t s(, + j) j b = a + a β t = j=! (3) u s(, + j) j = t (33)! j= Whe is fiite, u = sice it is a fiite sum of elemets of W ( ) The Stirlig umbers of the first id obey the recurrece relatio s(, j) = s(, j ) ( ) s(, j) Startig with s (,) =, a easy iductio argumet shows that s (, j)! for all ad j So whe is ifiite, we have u = j= j j= = t t j= j= s(, + j) t! t j j j < t (34) t j Lemma 3 applies to the last sum of this equatio givig us t = t j= j Therefore, t t = j= Sice this expressio is greater tha u, we must have u Substitutig u ito equatio (3) ad factorig a t out of the summatio, we have 7
Sice u for all, we ow t t β b = a + u = a (35) β u = a = = β a β a u (36) Sice β < ad a =, we have β a < So Lemma 3 applies to this sum ad we obtai β a = β a Thus, ( a ) u = β β a< = Callig this sum ξ, we ow have b t a t ( ξ ) a t a t t t = + = + ξ where aξ < t t a t a So a = b ad the theorem is prove The simplest applicatio of Theorem 36 is to the evaluatio of x ω lim x ( + cx) Whe we substitute ω for x, we have ( + cα ) Sice cα c e = + cα + c α! +, we have c α α + α = e Therefore, by Theorem 36 c c ad sice e αω = e, we have ( ) ω e c c + c α = e, which is equivalet to ω c ( + c α ) = e This happes idepedetly of our choice of ω, so x c lim x ( + cx) = e Theorem 36 tells us much more tha this I fact, it is ow easy to show that for ay f ( x ) satisfyig * f ( ω ) < x α, the limit lim x ( + cx+ f( x) ) does ot x differ from lim x ( + cx) We ca also use Theorem 36 to more easily evaluate limits of this form where c is a fuctio of x istead of a costat This is demostrated i the first example below Example 37 lim x l x ( + x ) x + x Solutio We first evaluate at ω to obtai the expressio ( + αlω) ω + Sice lω zoe( ), local equality o the level of α is equivalet to local α αlω α equality o the level of α lω So + α lω = e = ω, from which Theo- ω 8
ω ω rem 36 tells us that ( + α lω ) = ω We also have for the deomiator ω+ = ω ω, so by Theorem 33, ω ( + αlω) = ω + Therefore, the value of the limit is a Example 38 lim x 0 x + b x Solutio Evaluatig at α, we have Usig the idetity c x a α + b α α αl c = e, this ca be rewritte as e αl a + e ω αl b Addig the power series for the expoetials together, we obtai α( la lb) α (( la) ( l b + + + + ) ) + 4 Eve though the above expressio is ot the power series for ( l a+ l b) e = ab, it is locally equal to this power series o the level of α So we ca apply Theorem 36 to obtai ab α( la lb) α (( la) ( l b + + + + ) ) + = ab 4 Therefore, the value of the limit is ab Example 39 lim( cos x) x 0 cot x Solutio Evaluatig at α ad substitutig power series, we have ω ω ω 9
α! + 3 cot α α α α 3! + ( cosα ) = +! α! + For the expoet we ca use Theorem 33 to obtai 3 = ω sice α α 3! + cosα = ad siα = α So we write t = cot α = ω + β where β < ω Now for the base we have cosα = α = cot α t All of this shows that ( cosα ) satisfies the coditios for Theorem 36 Sice e α = α cosα ad cosα =, Theorem 36 tells us that cot α α ( cosα ) = ( e ) α = ( e ) = e cot + α β Sice β < ω, α β is a ifiitesimal which we will call γ We ow have e ω β α α β e e = e e γ = e + γ =! Usig Lemma 3, So we fially have e Therefore, the value of the limit is γ γ!! = = < = = γ γ γ + = e =! e 30
These examples would require a great deal more wor if we were to use l Hôpital s rule to evaluate the limits However, to oe who is adept at usig Theorems 33 ad 36, these limits ca be evaluated with far less effort 34 Ucompesated Square Completio Aother iterestig situatio arises for some limits which assume the form The example followig the ext theorem shows that we may sometimes add a umber to a expressio i order to complete a square without ever subtractig the umber elsewhere that is, we ever have to compesate for the chage to the expressio Theorem 30 Let t be a positive ifiite umber ad let c< t The t+ c = t Proof Expadig t + c with the biomial theorem we have Sice ( ) + = 0 = c = t + t (37) = t t c t c < for all, we ca write c c = (38) t = t Sice c< t, ct is a ifiitesimal So Lemma 3 applies to this sum givig c t = c t We ow have us ( ) = t = c t c (39) t Sice c< t, c t < Therefore, from equatio (37), t+ c = t + ξ where ξ < So t+ c = t Example 3 lim x + 6x x x 3
Solutio Evaluatig at ω, we have ω + 6ω ω Theorem 30 tells us that for ay c < ω + 6ω, we must have ω + 6ω+ c = ω + 6ω So we choose the oly value of c that is of ay advatage the oe which completes the square uder the radical Settig c = 9, we have Therefore, the value of the limit is 3 ω + 6ω ω = ω + 6ω+ 9 ω = ( ω+ 3) ω = 3 This method of ucompesated square completio provides a much faster alterative to the stadard method of multiplyig the expressio by its cojugate 3
4 Ifiite Series I this sectio we preset two ew tests for the covergece of ifiite series which are aalogs of the compariso test ad the limit compariso test of stadard aalysis Oce these tests have bee itroduced, we examie some useful facts which allow easier applicatio of the tests ad preset some examples 4 Covergece Tests Both of the ew tests determie the covergece of a series f ( ) by examiig the properties of * f ( ω ) The first test checs to see whether = * g( ω ) < * f ( ω ) for some coverget series f ( ) = Theorem 4 (Order Compariso Test) Let f ( x ) ad g ( x ) be stadard positive-valued fuctios If the series f ( ) coverges ad = * g( ω ) < * f ( ω ) (idepedet o the choice of ω) the the series g ( ) = also coverges Proof Sice * g( ω ) < * f ( ω ), we ow that * g( ω ) * f ( ω ) < ad thus * g( ω ) * f ( ω ) is a ifiitesimal Sice this happes idepedetly of the choice of ω, this implies that lim x g( x) f ( x) = 0 Therefore, give ay ε > 0, there exists a N N such that for all > N, g( ) f ( ) < ε Thus for all > N, g ( ) < ε f ( ) Sice ε f ( ) coverges, ( ) = g also coverges by the compariso = test Note that the cotrapositive of this theorem states that if the series g ( ) = diverges ad * g( ω ) < * f ( ω ), the the series f ( ) also diverges = Let z = ord (* g( ω )) for some stadard positive-valued fuctio g It immediately follows from the order compariso test that if z the the series p g ( ) coverges sice i this case we would have * g ( ω) < ω for some = real umber p with < p < z If z, the the series g ( ) diverges sice = p i this case we would have * g ( ω) > ω for some real p with z< p< For the remaiig case, z, we caot coclude aythig from the order compariso test This meas that if * f ( ω) = * g( ω), the whether f ( ) = 33
coverges tells us othig about whether g ( ) coverges However, if = * f ( ω) * f ( ω) = * g( ω) the we ca ifer the covergece of oe series from the other For this situatio, we have the followig test Theorem 4 (Local Equality Test) If f ad g are stadard positive-valued * f ( ω) fuctios that satisfy * f ( ω) = * g( ω) (idepedet o the choice of ω), the the series f ( ) ad ( ) = g either both coverge or both diverge = * f ( ω) Proof Sice * f ( ω) = * g( ω), we ow that * g( ω) * f ( ω ) =, which implies that * g( ω ) * f ( ω ) Sice this property is idepedet of the choice of ω, this meas that lim x g( x) f ( x) = Therefore, by the limit compariso test, the series f ( ) ad ( ) = g either both coverge or = both diverge 4 Usig the New Tests It is usually more coveiet to thi of a ifiite series as a sum of the reciprocals of a fuctio evaluated at each atural umber Fortuately, the order compariso test ad local equality test wor equally well for this situatio This is because * f ( ω) > * g( ω) implies * f ( ω) < * f ( ω) * g( ω) ad * f ( ω) = * g( ω) * f ( ω) implies * f ( ω) = * g( ω) I may cases whe the order compariso test caot be used, it will be possible through local equality to reduce the umber * f ( ω ) to a umber of the form aω where a zoe( ) Whe this happes, the followig extesio to the ( ) p-series test is useful We use the otatio l to mea the atural logarithm tae times (eg, l x = l l l x) ( ) ( 3) Let N ad let m be the least atural umber for which l m is real ad greater tha The the series ( ) ( ) = ml l l ( ) coverges if ad oly if p > We show this by usig the itegral test The itegral p 34
m ( ) ( ) xl xl x l x ( ) ca be evaluated by maig the substitutio u = l x for which we have ( ) ( ) du xlx l x l = x This gives us the itegral ( ) l m du, p u which coverges if ad oly if p > We coclude this sectio with a few examples Example 43 3+ 3 = 4 + 6 Solutio Let f 3 ( 4 6 ) = + We substitute ω for ad otice 3+ 3 4ω 6ω + = ω ω 3ω+ 3 4 6 So if we ca fid a fuctio g ( ) such that *g( ω ω ω ω ) = + for which ω 3ω+ we ow whether g ( ) coverges, the we ca use the local equality = test to determie whether f ( ) coverges By Theorem 33, = 3 4ω + 6ω ω 4ω = ω 3ω+ 3 3 ω 3 sice 4ω + 6ω = 4ω ad ω 3ω ω + = ω Because 4 diverges, the local equality test tells us that the series that we are testig also = diverges The ext example demostrates the procedure for dealig with factorials As show, the Stirlig approximatio for the factorial gives a good represetatio of the size of ω! Example 44 5 =! Solutio From Stirlig's approximatio to!, we ow that p dx ( ) 35