sign in sign up
<<
Home , Abraham Robinson, Hyperreal number, Real number, Transfer principle

Hyperreal Structures Arising From An Infinite

by

Eric Lengyel

Copyright © 1996

Abstract

This paper presents new concepts in the use of infinite and in . The theory is based upon the hyperreal system devel- oped by in the 1960's in his invention of “nonstandard analy- sis”. The paper begins with a short exposition of the construction of the system and the fundamental results of which are used throughout the paper. The new theory which is built upon this foundation organizes the of hyperreal numbers through structures which depend on an infinite base logarithm. Several new relations are introduced whose properties enable the simplification of calculations involving infinite and infinitesimal numbers. The paper explores two of application of these results to standard problems in elementary . The first is to the evaluation of limits which assume certain indeterminate forms. The second is to the determination of con- vergence of infinite . Both applications provide methods which greatly re- duce the amount of computation necessary in many situations.

1 Overview

n the 1960's, Abraham Robinson developed what is called “nonstandard analysis”, and in doing so provided a rigorous foundation for the use of infi- I nitesimals in analysis. A new number system known as the set of hyperreal numbers was constructed which includes the set of real numbers and also con- tains infinite and infinitesimal numbers. This paper begins with the construction of the hyperreals as a set of equivalence classes of of real numbers. Included in this introductory section is the method by which relations defined on the real numbers are extended to relations on the hyperreal numbers. Through these extensions, it is possible to prove statements that hold true over the hyper- real numbers which are the “nonstandard” equivalents of statements that hold true over the real numbers. In nonstandard analysis, the proofs of these state- ments are usually facilitated by the utilization of what is called the transfer prin- ciple. This concept, however, requires a great deal of development of rigorous which will not be needed in the of the paper. Therefore, the is not used in the introductory section at all, and alternate proofs of nonstandard results are instead given. In Section 2, we begin the study of new hyperreal structures which organize the set of hyperreal numbers into classes called zones, and we introduce the no- tions of superiority and local . These concepts depend on the of the hyperreal numbers taken to a fixed infinite base. We will be using infinite and infinitesimal numbers in such a way that our calculations will not require knowledge of how the hyperreal number system was constructed. Nor will our calculations require us to know what specific member of the set of hyperreals that we are using as the infinite base of our logarithm—knowing only that the number is infinite will suffice. Once the theory has been developed in Section 2, we proceed in Sections 3 and 4 to present applications in two areas. The first application is to the evalua- tion of certain limits which assume the indeterminate forms 00, ∞∞, 1∞, and ∞−∞. The methods presented will provide alternatives to l’Hôpital’s rule which generally allow much more efficient computation. The second application is to the determination of the convergence of infinite series. Two new will be presented which are analogous to the comparison test and com- parison test. As with limit evaluation, these tests significantly reduce the amount of computation in many situations.

2 1 Introduction to Nonstandard Analysis

his section presents the reader with the fundamentals of nonstandard analysis which are used throughout the remainder of this paper. We begin T with the ultrapower construction of the set of hyperreal numbers and then proceed to introduce several relations and algebraic structures which are defined on this set. This section consists only of the necessary background information which can be found in any introductory text on nonstandard analysis, most nota- bly [3]. Many of the proofs included in this section can also be found in the lite- rature, the exceptions being those which rely on the transfer principle.

1.1 Construction of the Hyperreal Number System The goals of the construction of the hyperreal number system are to build a which contains an isomorphic copy of the real numbers as a proper subfield and also contains infinite and infinitesimal numbers. Furthermore, it is desired that the numbers in this new field obey all of the same laws which hold true over the real numbers. A field having these properties is constructed by using a free ultra- to partition the set of all sequences of real numbers into equivalence classes. It is then these equivalence classes which are the elements of the set of hyperreal numbers. Before presenting the actual construction of the hyperreals, we include the of a filter.

Definition 1.1. A filter F on a set S is a nonempty collection of of S having the following properties.

(a) ∅∉F

(b) If A∈F and B ∈F then AB∩ ∈ F

(c) If A∈F and A ⊆⊆BS then B ∈F

Note that by (c), a filter F on S always contains S, and that by (a) and (b), no two elements of F are disjoint.

3 An on an S is a maximal filter on S. The existence of an ultrafilter follows from Zorn’s Lemma. A filter F on S is an ultrafilter if and only if it has the following property.

(d) If A ⊆ S then either A∈F or SA− ∈ F

An ultrafilter U on an infinite set S is called fixed or principal if there exists aS∈ such that U =⊆{A Sa| ∈ A}. which are not fixed are called free. An important fact is that free ultrafilters cannot contain any finite sets. By (d), this implies that if U is free then every cofinite of S is contained in U. We now construct the set of hyperreal numbers and prove that it is a linearly . We begin the construction by choosing a free ultrafilter U on the set of natural numbers N. The ultrafilter U is not explicitly defined since it does not matter which free ultrafilter on N that we use. The set of all free ultrafilters on N determines a set of isomorphic fields from which we can choose any mem- ber to be the set of hyperreal numbers. Using U, the hyperreals are constructed by considering the set of all sequences of real numbers indexed by N and defining the following on this set.

Definition 1.2. Given two sequences of real numbers an and bn ,

abnn≡ if and only if {nab∈ N | nn=∈} U . The entries of the sequences

an and bn are then said to be equal “ everywhere”.

The phrase “” (abbreviated a.e.) is used to signify that the entries of a have a certain property on some set in the ultrafilter U. For instance, if the sequence an has the property that an is an for all n in some member of U, then an is said to be an integer almost everywhere.

Proposition 1.3. The relation ≡ is an equivalence relation.

Proof. Let an , bn , and cn be sequences of real numbers. aann≡ , and thus ≡ is reflexive, since N ∈ U . Because equality on the reals is symmetric,

if abnn≡ then bann≡ , and thus ≡ is symmetric. For transitivity, sup-

pose abnn≡ and bcnn≡ . Let A = {nab∈=N | nn} and

B =∈{nbcN | nn =}. Then AB∩ = {nac∈ N | nn= }. Since A and B are ele-

ments of the ultrafilter U, AB∩ ∈ U . Therefore, acnn≡ .

The set of hyperreal numbers, which is denoted by *R, is defined to be the set of all equivalence classes induced by the equivalence relation ≡. We will use the notation [ an ] to represent the containing an .

4 , , and an ordering are defined on *R as follows.

Definition 1.4. Let [ abnn],*[ ]∈ R. The operations + (addition) and ⋅ (multiplication) and the relation < (less than) are defined by

(a) [ ababnn] +=+[][ nn]

(b) [ abnn]⋅=⋅[][ ab nn]

(c) [ abnn] < [] if and only if {nab∈ N | nn<∈} U Proposition 1.5. The operations + and ⋅, and the relation < are well-defined.

Proof. Suppose [ abnn] = [ ] and [ cdnn] = [ ], and let A = {nab∈=N | nn}

and Cn=∈{ N | cdnn =} (thus AC, ∈ U ). Then

{nacbdAC∈+=+=∩∈N | nnn n} U . So the two sums are equal elements of the hyperreals. The same argument holds true for multiplication.

Now suppose [ acnn] < [ ] and let Kn= { ∈<∈N | acnn} U . Then the

set on which bcnn< contains KA∩ and is therefore in U. So [ bcnn] < [].

Let Ln=∈{ N | bcnn <}. Then the set on which bdnn< contains LC∩ and is

therefore in U. Thus [ bdnn] < [ ].

We now claim that the set of hyperreal numbers combined with the opera- tions and ordering given in definition 1.4 is a linearly ordered field.

Proposition 1.6. The structure (*,,,R + ⋅<) is a linearly ordered field.

Proof. In order to show that *R is a field, we need only prove that *R con- tains inverses since the associative, commutative, and distributive laws as

well as are inherited from the real numbers. Let [ an ]∈*R such that

[ an ] ≠ []0,0,0,… . Then the set {na∈ N |0n =∉} U . Thus we know that the

of this set is in U. So define bn by 0, if a = 0; ⎪⎧ n bn = ⎨ ⎩⎪1,ifaann ≠ 0.

Then the [ abnn][ ] is equivalent to 1 since {nab∈==N |1nn }

{na∈≠∈N |0n } U . Therefore an is invertible and *R is a field.

Now let [ abnn],*[ ]∈ R such that [ abnn] ≠ [ ]. To prove that *R is

linearly ordered, we must show that either [ abnn] < [ ] or [ bann] < [].

Let A =∈{nabN | nn <}, B = {nba∈

5 [ abnn] ≠ [], E ∉ U . So the complement of E, which is AB∪ , is in U. If

A∈ U , then [ abnn] < [ ]. If A∉ U , then B ∪∈E U , in which case

( AB∪∩∪=∈) ( BE) BU , so [ bann] < [ ]. Thus [ an ] and [ bn ] are or- dered.

Now that we have shown that *R is a linearly ordered field, we wish to show that *R has a proper subfield which is isomorphic to R. We embed the reals in the hyperreals by defining the map θ :*RR→ by

θ (r) = [ rrr,,,… ].

To show that θ maps R to a proper subfield of *R, note that [ 1, 2, 3,… ] is not equivalent to the of any . This equivalence class is actually an example of an infinite number as is defined below.

1.2 Infinite and Infinitesimal Numbers Whether a number is infinite or infinitesimal is independent of the number's , so the of infinite and infinitesimal numbers will involve absolute val- ues. is a that is already defined on the real numbers which we need to extend to the hyperreal numbers. To do this, we simply apply the absolute value entrywise to an equivalence class representative in *R. Much more will be said about extending functions from the real numbers to the hyper- real numbers shortly, but right now we only need the following definition.

Definition 1.7. For all [ an ]∈*R, [ aann] = [ ].

Note that this definition is equivalent to the hyperreal analog of the definition of absolute value on the reals,

aa,if ≥ 0; ⎪⎧[ nn] [ ] []an = ⎨ ⎩⎪−[]aann,if []< 0.

This is because if [ an ] ≥ 0 then aann= on the same set in the ultrafilter for which an ≥ 0, and if [ an ] < 0 then −aann= on the same set in the ultrafilter for which an < 0. From this point on, we will use single letters to denote elements of *R. When we speak of a real number r ∈ *R, we mean the equivalence class [ rrr,,,… ]. Although technically speaking, the sets R, Q, Z, and N are not true subsets of the

6 hyperreals, we will use these symbols to refer to the images of these sets embed- ded in *R. We can now define what is means for a number to be infinite or infinitesimal.

Definition 1.8. Let a ∈*R. Then a is infinite if and only if ar> for every positive real number r. If a is not infinite then it is finite. We call a infinite- simal if and only if ar< for every positive real number r.

Note that the set of finite numbers is a of *R and the set of is an of the finite numbers. More importantly, note that the set of hyperreal numbers is nonarchimedean. Not every bounded subset of *R is guaranteed to have a least upper bound or greatest lower bound. For example, the set of infinite numbers is bounded below by any finite number but has no greatest lower bound. We now introduce an equivalence relation which associates numbers which only differ by an infinitesimal.

Definition 1.9. Given xy,∈ *R, x and y are near or infinitely close if and on- ly if x − y is an infinitesimal. In this case, we write x y. The equivalence classes induced by are called monads. Thus, the monad about a number a ∈*R, written ma( ) is defined by ma( ) = { x∈*|R x a}.

*R is also partitioned into classes whose elements differ by finite amounts.

Definition 1.10. The galaxy about a number a ∈*R, written Ga( ), is defined by Ga( ) =∈{ x*R | x − a is finite}.

Using monads and galaxies, we can use m(0) to represent the set of all infi- nitesimal numbers and G (0) to represent the set of all finite numbers. As shown below, the set of real numbers is isomorphic to the quotient Gm(00) ( ). Numbers in *R which are not images of real numbers are called nonstan- dard. Nonstandard finite numbers are always infinitely close to exactly one real number, which is called its standard part.

Proposition 1.11. Let a be a finite hyperreal number. Then there exists a unique real number r such that ra .

Proof. Let A =∈{xxaR | ≤}. Since a is finite, A is nonempty and is bounded above. Let r be the least upper bound of A. For any real ε > 0, rA−∈ε and rA+∉ε and thus rar− ε ≤<+ε . So ra− ≤ ε from which is follows that ra . To show that this r is unique, suppose that there exists a

7 real number s such that s a. Then since is transitive, s r . So sr−<ε for every real ε > 0, and thus s = r.

Definition 1.12. Let aG∈ (0). Then the standard part of a, denoted by st (a) or °a, is the real number r such that ar . The function st is called the stan- dard part map.

The standard part map is easily shown to be an order preserving from G (0) onto R with m(0). The x − st ( x) is sometimes called the nonstandard part of x.

1.3 Relations and *-Transforms We now discuss the method through which functions defined on the set of real numbers are extended to the hyperreals. The method actually applies to any arbi- trary relation defined on R, the set of which includes all of the functions defined on R. The process of extending a relation from R to *R is called a *-transform for which the general definition is next given.

Definition 1.13. Let P be an n-ary relation on R. Then the *-transform of P,

denoted by *P is the set of all n- ([ (aa12)ii],,,[ ( ) ] … []() an i) n ∈()*R satisfying {iaaaP∈N |,,,(( 12)ii( ) … ( n ) i)∈∈} U . An n-ary relation on R is simply a subset of Rn. Thus, unary relations on R are just subsets of R. Equality, the ordering given by <, and functions of one va- riable are examples of binary relations. In order to acquire a more intuitive feel- ing for how relations on R are extended to *R, we consider a few examples. Let A ⊆ R. Then A is a unary relation on R, so *A consists of those elements

[ an ]∈*R such that {naA∈ N | n ∈∈} U (i.e., aAn ∈ almost everywhere). It should now be clear why the notation *R is used to represent the set of hyperreal numbers, for if A = R then {naA∈ NN| n ∈=∈} U , so *R consists of all equiva- lence classes represented by any sequence of real numbers. The set *N is called the set of hypernatural numbers and consists of the numbers [ an ]∈*R for which an ∈ N almost everywhere. Likewise,

**|ZRZ=∈∈{[]aann a.e.} and **|QRQ= {[ aann]∈∈ a.e.}. These are called the and hyperrationals respectively.

Proposition 1.14. Let S ⊆ R be a set which contains an infinite subset of N. Then *S contains infinite numbers.

8 Proof. Let A be an infinite subset of N contained in S. We construct an infi-

nite number in *S as follows. Let a1 be the least element of A and then

choose each an to be the least element of the set A −{aa12,,,… an− 1}. Since A

has no upper bound, the sequence an must have the property that given any positive real number r, there exists an N ∈ N such that for all nN> , we have

arn > . Thus arn > almost everywhere, satisfying the requirement for

[ aSn ]∈* to be infinite.

Given a set S ⊆ *R, we use the notation S∞ to represent the set of infinite numbers in S. Thus, *Z∞ is the set of infinite hyperintegers in *R. Let us consider equality on R for a as a set of duplets and write (ab, )∈ E if and only if ab= . Then *E is the set of all duplets 2 ()[]abnn,*[]∈()R satisfying {nab∈ N | nn=∈} U , which is exactly how we defined equality on the hyperreals. A function f on R of n variables can be thought of as a set of (n +1)-tuples having the property that if (cc12,,,,… cn a)∈ f and (cc12,,,,… cbn )∈ f, then ab= . The *-transform of f is the set of all (n +1)-tuples of the form n+1 ()[]()cc12ii,,,,[]() … [ () cann i] []∈() *R satisfying

{icccaf∈N |,,,,(( 12)ii( ) … ( ni) i)∈∈} U .

So if f is a function of one , then * f ([ cii]) = [ f (c ) ]. That is, we just let f operate entrywise on an equivalence class representative of a number in *R. Note that operations such as addition and multiplication are actually func- tions of two variables. In the next section, we will be taking logarithms of hyper- real numbers using a hyperreal number for the base. The logarithm is also a function of two variables so we have

*log[]aa= ⎡ log⎤ . []bn nbn⎣ n ⎦

If we choose [ bn ] = [ 1, 2, 3,… ] and [ an ] = [ 2,2,2,… ], then the first entry of *log a is log 2, which is undefined. This is acceptable, however, since []bn []n 1 we can assign any value we wish to the first entry and the function will still hold true on a set in the ultrafilter U. As long as the *-transform of a function is de- fined almost everywhere for the entries of an equivalence class representative

an , it is defined for [ an ].

9 1.4 Sequences and Series Sequences and series of hyperreal numbers will be an important of study later in this paper. When we speak of a sequence of hyperreal numbers, we are actually talking about a sequence of sequences of real numbers, and this se- quence is indexed not by the natural numbers, but by the hypernatural numbers. Below, we examine how sequences of real numbers indexed by the natural num- bers are *-transformed to sequences of hyperreal numbers indexed by the hyper- natural numbers.

Let snn | ∈ N be a sequence of real numbers. This sequence is actually a function s : NR→ where s(ns) = n , so we can think of sn as the set of duplets

{(1,sss123) ,( 2,) ,( 3,) ,…}. The *-transform of s is the function *:*s NR→ * where *s()[]asann= []( ) for any [ an ]∈*N. The sequence *|snn ∈ *N is an extension of the sequence snn | ∈ N in that for any n∈N, *sn is simply the im- age of sn in the hyperreals.

If a sequence sn tends to a limit in the real numbers, then the sequence

*sn tends to the same limit in the hyperreal numbers. This is proven shortly, but first we need to discuss a little notation. The symbol ∞ is used in the real number system to denote that which is potentially arbitrarily large. Thus the expression limnn→∞ s denotes the value that sn approaches as n becomes an arbitrarily large . In the hyperreal number system, the symbol ∞ has the same meaning, but by arbitrarily large, we mean even larger than any infinite number in *R. So the expression limnn→∞ * s denotes the value that *sn approaches as n becomes an arbitrarily large hypernatural number. We now have the following proposition.

Proposition 1.15. Suppose that limnn→∞ s = L for some real number L. Then

limnn→∞ * s = L.

Proof. The fact that limnn→∞ s = L means that given ε > 0 there exists an

N ∈N such that for all nN> , sLn − < ε . Now let ε be a positive hyperreal

number and let ε i be an equivalence class representative of ε. We define

the hypernatural number NN= [ i ] by choosing Ni to be any natural number

for which sLni−<ε for all nN> i . Then for any hypernatural number nN> ,

*sLn −<ε . So limnn→∞ * s = L.

More important to this paper are infinite series of hyperreal numbers. We define an infinite series in terms of its sequence of partial sums. Consider the standard sequence

10 n s()()nx,,,12 x…… , xmm= ∑ akx ,,, 12 x , x , k =1 which is defined as a function of the natural index n and real numbers x12,,,xx… m. The sum of the infinite series is represented by limnm→∞ s(nx ,12 , x ,… , x ). The *-transform of s is the sequence

n *s()()nx ,,,,12 x…… xmm= **∑ akx ,,,, 12 x x , k =1

where x12,,,xx… m are hyperreal numbers and the right hand side represents the definition of the nonstandard summation from 1 to a hypernatural number n. This summation is written in equivalence class form as

n ⎡ ni ⎤ …… **∑∑akx()()()() ,,,,12 x xmm= ⎢ ak() , x 1ii , x 2 ,, x i⎥ . kk==11⎣ ⎦

An important addition to this definition is the requirement that for each of the hyperreal numbers x12,,,xx… m, only one equivalence class representative may be used throughout the entire summation. That is, to calculate s(nx,,,12 x… , xm ) for a single hypernatural number n, we first choose equivalence class representatives for each of the x12,,,xx… m and use the same representatives each that we evaluate *,,,,akx( 12 x… xm ) in the summation. Without this restriction, is it poss- ible to find ill-defined summations when n is infinite by choosing different equi- valence class representatives for each index k. The sum of the infinite series

n **∑ akx() ,,,,12 x… xm k =1

is defined to be limnm→∞ *s(nx ,12 , x ,… , x ) where n approaches through the hypernatural numbers. If this limit exists and is equal to any hyperreal num- ber, including infinite numbers, then the series is said to be convergent. We consider an example that is used later in this paper. Let s(nabt,,,) represent the sequence of partial sums of the binomial series for ()ab+ t . This can be written as the summation

n ⎛⎞t s nabt,,,= atk− b k . ()∑⎜⎟ k =1 ⎝⎠k

11 In order to use the binomial expansion in the case where a, b, and t are hyperreal numbers, we need the *-transform of s, which is given by

n ⎛⎞t *,,,*snabt= atk− b k ()∑⎜⎟ k =1 ⎝⎠k n ⎡ i ⎛⎞ti ⎤ = abtki − k . ⎢ ∑⎜⎟ii⎥ ⎣⎢ k =1 ⎝⎠k ⎦⎥

In this summation, we must use fixed pre-chosen equivalence class representa- tives for a, b, and t for every index k.

1.5 Limits One final result of nonstandard analysis that is needed in this paper is the follow- ing statement about the at a or at infinity.

Proposition 1.16. Let L be a real number.

(a) lim xa→ f ( xL) = if and only if * f ( xL) for all x ∈ ma( ) −{ a}.

(b) lim x→∞ f ( xL) = if and only if * f ( xL) for all positive infinite x. Proof. All parts are proven by contrapositive.

(a) Suppose * f ( zL) / for some za . We want to show that there exists an ε > 0 such that for any δ > 0 there exists an x such that xa−<δ , but fx( ) −≥ L ε . Since * f ( zL) / , we have *0fz( ) − L . If * f ( z) is fi- 1 nite, choose ε = st( 2 * f ( zL) − ), and if * f ( z) is infinite, choose ε to be

any positive real number. Let zn be an equivalence class representative of z. Since za , we know that

Zn= { ∈−<∈N | zan δ } U .

We also know that

Fn= { ∈−≥∈N | fzL( n ) ε} U .

Choose NZ∈∩ F and let x = zN . Then xa− < δ , but fx( ) −≥ L ε .

Therefore lim xa→ f ( xL) ≠ .

12 Now we assume lim xa→ f ( xL) ≠ . Then there exists an ε > 0 such that for every δ > 0 we can find x such that xa−<δ , but

fx( ) −≥ L ε . We construct a sequence xn by choosing each xn such

that xn −

any positive real number r, xn − ar< for all nN> where N is the least

natural number satisfying 1 Nr< . But * f ([ xLn ]) / because

fx( n ) −≥ L ε for all n∈N. So we have found a number infinitely close to a for which the function value is not infinitely close to L. (b) Suppose * f ( zL) / for some positive infinite number z. We want to show that there exists an ε > 0 such that for any m ∈ N there exists an x > m such that fx( ) − L≥ ε . As in part (a), *0fz( ) − L . So if * f ( z) 1 is finite, choose ε = st( 2 * f ( zL) − ), and if * f ( z) is infinite, choose ε to

be any positive real number. Let zn be an equivalence class representa- tive of z. Since z is infinite, we know that

Zn= { ∈>∈N | zmn } U .

We also know that

Fn= { ∈−≥∈N | fzL( n ) ε} U .

Choose NZ∈∩ F and let x = zN . Then x > m, but fx( ) −≥ L ε . There-

fore lim x→∞ f ( xL) ≠ .

Now we assume lim x→∞ f ( xL) ≠ . Then there exists an ε > 0 such that for every N ∈N we can find x > N such that fx( ) −≥ L ε . We

construct a sequence xn by choosing each xn such that xn > n and

fx( n ) −≥ L ε . Then [ xn ] is infinite since for any positive real number

r, xn > r for all nr> . But * f ([ xLn ]) / because fx( n ) −≥ L ε for all n∈N. So we have found an infinite number for which the function value is not infinitely close to L.

13 2 Development of New Hyperreal Structures

his section introduces the concept of superiority and the equivalence rela- tion called local equality. Many structures which arise in the set of hyper- T real numbers due to these concepts are examined along with their relationships to existing structures. Succeeding sections discuss applications of the properties of these new structures. The following notation is used in this and later sections. The set of all infinite numbers in *R will be denoted by T, and the set of all infinitesimal numbers in *R will be denoted by S. The symbols and will be used to mean “less than or infinitely close to” and “greater than or infinitely close to” respectively. The symbols and will be used to mean “less than but not infinitely close to” and “greater than but not infinitely close to” respectively.

2.1 Orders of Numbers and the Superiority Relation The order of a number and all subsequently dependent relations are based upon the choice of a positive infinite number ω. We do not give an explicit equiva- lence class representative for ω since all calculations that we perform which in- volve ω do not actually rely on its value. Thus all of the relations and structures defined in this section have the same properties for every possible choice of ω. We need only remember that during any one calculation that ω represents a con- stant positive infinite hyperreal number. This method allows us to perform com- putations in the hyperreal numbers and at the same time abstract beyond the need to know anything about the ultrafilter used to construct the hyperreals or the se- quences that represent equivalence classes. All symbolic manipulation is done with real numbers and functions of ω. We use the letter α to represent 1 ω for any choice of ω. The arbitrary nature of ω and α allows us to show in a single calculation that certain properties hold for all positive infinite or infinitesimal numbers. We first define an equivalence relation that partitions *R into what are called zones. This is done by considering the logarithm base ω of each hyperreal num- ber. We will use the symbol log to represent the function *log on *R as well as the function log on R.

14 Definition 2.1. The order of a number a ∈*R, written ord(a), is defined as

ord(aa) = logω . The order of 0 is defined to be −∞.

The order of a number represents the size of the number on a scale which tran- scends infinitesimal, finite, and infinite numbers. Although this order depends upon the choice of ω, we have the following property that does not depend on the value of ω, but only on the fact that ω is infinite.

Lemma 2.2. All numbers in Gm(00) − ( ) (i.e., all finite non-infinitesimals) have infinitesimal order.

Proof. Let xG∈−(00) m( ). Since x is not infinitesimal or infinite, there ex- ist a and b such that axb< < . For every positive real number r, ord(rr) = ln lnω ∈S. Since the logarithm is a strictly increasing function, we have ord(axb) < ord( ) < ord( ). So x has infinitesimal order.

Clearly, every infinite number has positive order and every infinitesimal number has negative order. What is interesting is that although most infinite and infinitesimal numbers do not have infinitesimal order, there are elements of T and S that do, and these numbers are given the following special names.

Definition 2.3. If t ∈T and ord(t)∈S, then t is called semi-infinite. The set of infinite numbers less the set of semi-infinite numbers is denoted by T′.

Definition 2.4. If s ∈S and ord(s)∈S, then s is called semi-infinitesimal. The set of infinitesimal numbers less the set of semi-infinitesimal numbers is denoted by S′.

An example of a semi-infinite number is lnω whose order is ln lnω lnω. All semi-infinitesimal numbers are reciprocals of semi-infinite numbers and vice- versa, so 1lnω is semi-infinitesimal. Properties of semi-infinite and semi- infinitesimal numbers are identified throughout this section. The order of each hyperreal number is used to define the following relation on *R. Definition 2.5. Given ab,*∈ R, we say a and b are isometric and write ab= if and only if ord(ab) ord( ). The relation = is an equivalence relation since S is a under addition. The name “isometric” is used because two numbers whose orders differ only by an

15 infinitesimal are of the same general size when considered as members of the vast infinitesimal and infinite extent of the set of hyperreal numbers. We now introduce the superiority relation that is used to relate two numbers which are not isometric. Definition 2.6. Let ab,*∈ R such that ab=/ . If ord(ab)() ord , then we say a is inferior to b and write ab< , and if ord(ab) ord( ), then we say a is su- perior to b and write ab> . The symbol ≤ is used to mean “inferior to or isometric to”, and the symbol ≥ is used to mean “superior to or isometric to”. Note that 0 < a for all nonzero a ∈*R. Also note that if ab≤ then ab≤ . This property will be useful in several proofs later in this paper. If two numbers a and b in *R are related by =, <, or >, then multiplication of both by any nonzero x ∈*R preserves this relation. Lemma 2.7. Let abx,,∈ *R. Then ab= implies ax= bx, and if x ≠ 0 then ab< implies ax< bx. Proof. Suppose ab= . Then ord(ab) ord( ). Adding ord( x) to both sides, we have ord(axbx) ++ ord( ) ord( ) ord( ). Therefore ord(ax) ord(bx), so ax= bx. Now suppose ab< . Then ord(ab) ord( ), so by a similar argument ord(ax) ord( bx) and thus ax< bx. Addition of equals does not preserve the relations =, <, and >. Consider as an 2 example that 11+=+α α , but if we add − 1 to both sides then we would have 2 aa= , which is not true.

2.2 Zones and Worlds An equivalence class created by the equivalence relation = is given the following name. Definition 2.8. The equivalence class induced by = containing a ∈*R is called the zone about a. This is written zone(ax) = { ∈= *R | xa}. We define zone( 0) = { 0}.

The zone about a is the set of all numbers whose order is in the monad about ord(a). Thus it is possible to express the zone about a as

16 zone(a) =∪−ωωma(ord( )) ma( ord( )) .

Note that since ord( x) = ord(−x) for all x ∈*R, if x ∈ zone(a) then −∈x zone(a). Also note that since ord( 1) = 0, zone( 1) is the set of all elements of *R having infinitesimal order. Therefore zone( 1) contains all of the nonzero real numbers.

Theorem 2.9. zone( 1) is a multiplicative subgroup of *R.

Proof. We need to prove closure and containment of inverses. Let ab,zone1∈ ( ). Then ord(a)∈S and ord(b)∈S. Therefore, ord(ab) = ord(ab) +∈ ord( ) S. So ab∈ zone(1) and thus zone( 1) is closed under mul- tiplication. zone( 1) contains multiplicative inverses since for all x ∈*R, ord( x−1 ) =− ord( x) and S contains additive inverses. Every zone is a coset of zone( 1). We therefore have the properties that for all a ∈*R, zone(aa) = zone( 1) and zone(a) is closed under multiplication by ele- ments of zone( 1). All semi-infinite and semi-infinitesimal numbers are contained in zone( 1). We can therefore express the set of semi-infinite numbers as T ∩ zone( 1), and we can express the set of semi-infinitesimal numbers as S ∩ zone( 1). No zone (excluding zone( 0)) is closed under addition since for any ba< , ab+∈zone( a), but aba+ −∉zone( a). The smallest additive subgroup of *R containing zone(a) therefore contains every number which is inferior to a. This set is given the following name.

Definition 2.10. The world about a ∈*R, denoted Wa, is defined by ( ) Wa( ) =∈{ x*|R x ≤ a}. We use the notation Wa0 ( ) to represent the set {x ∈*|R xa< }. Wa0 ( ) can be thought of as the interior of the set of zones contained in Wa( ), and zone(a) can be thought of as the boundary of the set of zones contained in Wa( ). We thus have for the zone about a the alternate notation ∂Wa( ).

Theorem 2.11. Wa( ) and Wa0 ( ) are groups under addition for all a ∈*R. Proof. 0∈Wa( ) and 0∈Wa0 ( ) because 0 ≤ a for all a ∈*R. Each zone which is contained in Wa( ) contains its own additive inverses. We are left with proving closure, which we first prove for the Wa( ) case.

17 Let x, yWa∈ ( ). Without loss of generality, we can assume that xy≥ . We will show that x + yWx∈⊆( ) Wa( ). This means we must show that ord( x + yx) ord( ). Let zxy= max( , ). Then

ord( xy+ ) =+ logω xy

≤+logω ()x y

≤ logω 2z

=+logωω 2 log z ord()z . (2.1) If x = y then zx= . If x > y then zx= . In both cases, ord( zx) ord( ). Therefore, from the above , ord( x + yx) ord( ). Closure of Wa is proven in the exact same manner since for 0 ( ) x, yWa∈ 0 ( ) with xy≥ , Wx( ) ⊂ W0 ( a). An important observation is that if x > y then x + yx= since otherwise, if x +=

Wa( ) is thus closed under multiplication. For x, yWa∈ 0 ( ), we have the simpler situation that ord( x) < 0 and ord( y) < 0. Therefore, ord( xy) < 0 and we have xyWa∈ ( ). 0 Now suppose that a >1. Then ord(a) 1. So we have ord(a2 ) = 2ord()aa ord () and thus aWa2 ∉ ( ). Therefore Wa( ) is not closed under

multiplication. Also, Wa0 ( ) is not closed since if we let b = aWa∈ 0 ( ), 2 then bWa∉ 0 ().

It turns out that for any two rings of the mentioned in Theorem 2.12, the smaller subring is an ideal of the larger. Theorem 2.13. Let ab,*∈ R such that ab< ≤ 1. Then

(a) Wa( ) Wb( )

18 (b) Wa0 ( ) Wb( )

(c) Wa( ) W0 ( b)

(d) Wa00( ) Wb( ) Proof. (a) Let x ∈Wa( ) and yWb∈ ( ). Then ord( x) 0 (since a <1) and ord( y) 0. So ord( xyxy) = ord( ) + ord( ) ord( x), which implies xy ≤ x, and thus xyWa∈ ( ). (b) Identical to (a). (c) Similar to (a), except ord( y) 0. So ord( xy) ord( x), which implies xyx< . (d) Identical to (c).

Theorem 2.14. W0 (1) is a of W (1). Proof. Suppose there exists an ideal I of W (1) with WIW(11) ⊂⊂ ( ). Then 0 there exists x∈ I such that x =1 and is therefore an element of zone( 1). But by Theorem 2.9, zone( 1) contains multiplicative inverses, so x is a unit and

thus IW= (1). So W0 (1) must be maximal.

We have now constructed a field WW(11) 0 ( ) considerably smaller than *R that still contains an isomorphic copy of the real numbers through the map r…[] rrr,,,+ W0 ( 1) and also contains infinite and infinitesimal numbers such as lnω +W0 ( 1) and 1lnω +W0 ( 1).

2.3 Local Equality The concept of local equality involves choosing a “level” to work on and equat- ing any two numbers in *R that differ by an insignificant amount with respect to this level. For example, on the finite level (i.e., the level of 1), any two numbers that differ by a sufficiently small infinitesimal are considered to be locally equal. The precise definition is as follows.

Definition 2.15. Let ab,,∈ *R with ≠ 0. We say a is locally equal to b on the level of and write ab= if and only if ab− < .

19 Local equality on the level of is an equivalence relation since 0 < for all ∈*R and W () is closed under addition by Theorem 2.11. One immediate ob- 0 servation is that if x < then x = 0. If ab= , then obviously ac+= bc + for any c ∈*R. Also, it follows immediately from Lemma 2.7 that if ab= then ac=c bc. If two numbers are locally equal on the level of then they are also locally equal on the level of any element of zone(). Replacing the level representative with an isometric number does not change anything. Local equality is also pre- served if the level representative is replaced by a superior number. In general, if ab= and m ≥ , then ab=m .

a Lemma 2.16. If ab= then ab= . Proof. If ab=a then aba− < . So we can write ba= + ε where ε < a. Since aa+=ε , we know ba= .

An immediate consequence of this lemma is that if ab=a then ab=b . This also lets us make the statement that if ab=a then 11ab=1 a . This is because if we divide both sides of ab=a by a, we have 1=1 ba. Then dividing both sides by b gives us 11ba=1 b . By the lemma, this is equivalent to 11ab=1 a . This property of local equality is useful in Section 4. Here are some example local equalities.

α =12α 34=ω 56+=α 1 5 1 77+≠1 lnω

Note that in the last example given above, even though 1lnω is an infinitesimal, it is too large to be ignored on the finite level. The reason is that 1lnω is a semi- infinitesimal and semi-infinitesimals are not inferior to any finite number. The equivalence classes of local equality on the level of 1 are analogous to monads. Numbers that are in ma( ) but not in {x | xa=1 } are exactly those which differ from a by a semi-infinitesimal. So xy=1 implies x y, but the converse is not necessarily true.

20 2.4 Summary All of the structures and relations introduced in this section depend upon a posi- tive infinite number ω whose exact value is left undefined but is considered to be held constant throughout any statement, proof, calculation, etc. Whenever the number ω appears in an expression, it is assumed to be the same ω that any rela- tions appearing in the expression such as inferiority and local equality depend on through the order function. The arbitrary nature of ω allows us to conclude statements such as if f (ω) L independently of the choice of ω, then f (tL) for all positive infinite numbers t. This becomes an invaluable tool when used in conjunction with local equality for evaluating limits which assume an and also for testing infinite series for convergence. These topics are discussed in the next two sections.

21 3 Evaluation of Limits

e now use the concept of local equality to develop a system for eva- luating certain types of limits at points where they assume indetermi- Wnate forms. The methods presented are alternatives to l’Hôpital’s Rule which are generally easier to use and in most cases allow much faster com- putation. Several examples are given which demonstrate the new methods on limits which assume the indeterminate forms 00, ∞ ∞, 1∞, and ∞−∞.

3.1 Preliminary Theory

Nonstandard analysis tells us that lim x→∞ f ( x) exists and is equal to L if * f ( xL) for all positive infinite x ∈*R (see Section 1.5). When evaluating lim- its of this form, we will be able to show that * f (ω) =1 L independently of the choice of ω. Thus, in a single calculation, we will be able to show that * f ( xL) for every infinite number ω. This implies that lim x→∞ f ( xL) = .

A similar method will be used for limits of the form lim x→0 f ( x). In this case, we need to show that * f ( xL) for all infinitesimal x. We will be able to obtain * f ()α =1 L independently of the choice of α, from which is follows that lim x→0 f ( xL) = . The way in which we will use local equality at first is by using the fact that given px( )∈ R[] x, if we choose a∈ST′ ∪ ′, then every of px( ) evaluated at a belongs to a different zone. Thus we have properties such as p(ω) is locally equal on its own level to the highest degree term of px( ) evaluated at ω. The following lemma becomes useful in several places where we are work- ing with infinite series of infinitesimals. (The * is omitted from the summations in this section – all summations are nonstandard.)

∞ n Lemma 3.1. If β is an infinitesimal, then β = β . ∑ n=1 Proof. We have the equation

22 ∞∞ ∑∑ββnn≤ nn==11 ∞ = ββ∑ n−1 n=1 n−1 ∞ ⎛⎞1 < β ∑⎜⎟ n=1 ⎝⎠2 = 2 β = β . (3.1)

∞ n ∞ n This tells us that β ≤ β . Since β itself is a summand, β = β . ∑ n=1 ∑ n=1 In many of the examples in this section, we use to represent ex- ponential and . These series are then evaluated at α, mak- ing it possible to use the following lemma. Lemma 3.2. Suppose β < 1 and let ann | ∈ N be a sequence of real numbers

such that an is bounded by some real number m. Then for any finite k, ∞ k k **aaβ nn=β β . ∑∑nn==11nn ∞ Proof. We need to show that *a β nk< β . We can write this sum as ∑ nk=+1 n

∞∞ nn ∑∑**aannββ≤ nk=+11 nk =+ ∞ knk− = ββ∑ *an nk=+1 ∞ kn = β ∑ *ank+ β n=1 ∞ ≤ m ββkn∑ . (3.2) n=1

∞ n By Lemma 3.1, β = β . Therefore, ∑ n=1

∞ knk +1 m βββ∑ = . (3.3) n=1

This tells us that

23 ∞ nk +1 ∑ *.an β ≤ β (3.4) nk=+1 Since β < 1, β kk+1 < β , so the entire sum is inferior to β k.

3 This lemma allows us to make statements such as sinααα=−α 3 6.

3.2 Limits of the Form 00 and ∞ ∞

For limits which assume the form 00 or ∞ ∞, we need to examine the properties of local equality as it pertains to quotients ab. If ab< , then ab=1 0, and if ab> , then ab is infinite. For the remaining case, ab= , we have the following theorem.

Theorem 3.3. Let a1, a2 , b1, and b2 be nonzero elements of *R. Suppose we aa have the local equalities aa=a1 and bb=b1 . Then 12=ab11 . 12 12 bb12 Proof. Write aa= + ε where ε < a and write bb= + ε where ε < b . 21a a 1 21b b 1 Multiplying ε by b and ε by a , we have the relations babε < and a 1 b 1 111a aabε < . Since Wab( ) is closed under addition, ababε −<ε . By 111b 011 11ba 11 Lemma 2.16, bb12= . So we can write abab11ε ba− ε < 12. (3.5)

The left side of this equation can be rewritten by adding and subtracting ab11 to obtain ab11( + ε ba) −+< ba 1( 1ε ) ab 12. (3.6)

Replacing b1 + εb with b2 and a1 + ε a with a2 and then dividing both sides by

bb12 (using Lemma 2.7), we have ab− ab a 12 21< 1. (3.7) bb12 b 1

aa12 The left side of this equation is simply bb− , so the proof is complete since aa12 this is now the definition of 12=ab11 . bb12 A trivial example of the application of Theorem 3.3 is a quotient of polyno- mials. If we have lim x→∞ px( ) qx( ), then we substitute ω for x in both px( ) and

24 qx( ). If pq(ω ) = (ω ), which is true if and only if p and q have the same degree, then the limit is equal to the quotient of the leading of p and q. Fol- lowing are two slightly less trivial examples in which trigonometric functions appear.

sin x Example 3.4. lim x→∞ x + x2

Solution. We replace the function with its power series and evaluate at α to obtain

α − αα35+−+ 3! 5! . α +α 2

αα35 α 2 α By Lemma 3.2, α − 3!+−+= 5! α and α +=αα, so we can apply Theorem 3.3 as follows.

α −+−+αα35 α 3! 5! =1 =1. α + αα2

Since this local equality holds for any choice of α, we know limsin x = 1. x→0 xx+ 2 A similar example involving the cosine function is given next.

1cos− x Example 3.5. lim x→0 x24− 3x

Solution. As above, we replace the cosine function with its power series and evaluate at α to obtain

24 αα αα24 11−−( 2! + 4! −+) − +− = 2! 4! . α 24− 33ααα 24−

2 2 αα24 α 1 2 24α 2 This time we have 2!−+−= 4! 2 α and α −=3αα, so applying Theorem 3.3,

αα24−+− 1 α 2 1 2! 4! =1 2 = . α 24− 32αα 2

Again, this local equality is independent of α. So lim 1cos− x = 1 . x→0 xx24−3 2

25 Evaluating the limit in Example 3.5 would require two applications of l’Hôpital’s rule. However, using local equality enables the limit to be evaluated in a single step once the trigonometric function is replaced by its power series.

3.3 Limits of the Form 1∞ Far more interesting situations arise when we examine limits which assume the form 1∞. For this case, we need the following theorem. Theorem 3.6. Let a ∈*R and suppose at−11= for some positive infinite t number t ∈*R. Then for any b ∈*R satisfying ba=1 t , we have abta= t.

Proof. Let ε =−ba and assume ε ≠ 0. Expanding ()a + ε t with the binomial theorem, we obtain

∞ ⎛⎞t battkk=+ε t = a− ε . (3.8) ()∑⎜⎟ k =0 ⎝⎠k

For k = 0, the summand is simply at . We wish to show that the sum of the terms for k ≥1 is inferior to at . So we rewrite equation (3.8) as

∞ ⎛⎞t batt=+ a tkk− ε (3.9) ∑⎜⎟ k =1 ⎝⎠k

and show that this sum for k ≥1 satisfies the hypothesis of Lemma 3.1. (We will actually be applying this lemma twice.) The binomial t =−tt()()(12 t − t −+ k 1! ) k is a degree k ( k ) in t. When the numerator is expressed as a sum of powers of t, the coefficient of the degree j term is given by the Stirling number of the first kind s(kj, ) where skj( ,0) = if j < 1 or jk> . Thus we can write equation (3.9) as

∞ k tt⎛⎞ tkk− skj(), j ba=+∑∑⎜⎟ aε t. (3.10) kj==11⎝⎠k! Since ε <1 t, we can write ε = β t where β < 1. Replacing ε with β t and factoring t k out of the inner sum, we have

26 ∞ k tt⎛⎞ tkk−−skj(), jk ba=+∑∑⎜⎟ aβ t . (3.11) kj==11⎝⎠k!

We now reverse the order of summation for the inner sum through the map jk +−1 j so that the terms are arranged in the order necessary for the ap- plication of Lemma 3.1. This gives us

∞ k tt⎛+−⎞ tkk−−skk(),1 j 1 j ba=+∑∑⎜⎟ aβ t . (3.12) kj==11⎝⎠k!

For convenience, we define

k skk( ,1+ − j) 1− j utk = ∑ . (3.13) j=1 k! When k is finite, uk =1 since it is a finite sum of elements of W (1). The Stirl- ing numbers of the first kind obey the

s(kj,1,111,.) = sk( −−−− j) ( k) sk( − j)

Starting with s(1,1) = 1, an easy induction argument shows that s(kj,!) ≤ k for all k and j. So when k is infinite, we have

k skk( ,1+− j) 1− j utk = ∑ j=1 k! k ≤ ∑t1− j j=1 j k ⎛⎞1 = t∑⎜⎟ j=1 ⎝⎠t j ∞ ⎛⎞1 < t∑⎜⎟. (3.14) j=1 ⎝⎠t

∞ − j Lemma 3.1 applies to the last sum of this equation giving us tt=1 . ∑ j=1 ∞ − j Therefore, tt=1. Since this expression is greater than u , we must ∑ j=1 k t have uk ≤1. Substituting uk into equation (3.12) and factoring a out of the summation, we have

27 ⎛⎞∞ β k batt1. u (3.15) =+⎜⎟∑ k k ⎝⎠k =1 a Since uk ≤1 for all k, we know

∞∞ββkk uu≤ ∑∑kkkk kk==11aa ∞ β k ≤ . (3.16) ∑ k k =1 a Since β < 1 and a =1, we have β a <1. So Lemma 3.1 applies to this sum ∞ kk ∞ kk and we obtain β aa= β . Thus, (ββau) k ≤< a1. Calling ∑ k =1 ∑ k =1 t this sum ξ, we now have batt=+=+(1 ξ ) aa ttξ where aattξ < . So abta= t and the theorem is proven.

The simplest application of Theorem 3.6 is to the evaluation of x ω limx→∞ () 1+ cx . When we substitute ω for x, we have ()1+ cα . Since ecccα =+12!αα +22 +−, we have 1+=ceα α cα . Therefore, by Theorem 3.6 c and since eeccαω = , we have ()1+=ceα ω ec, which is equivalent to ()1+=ceα ω 1 c. This happens independently of our choice of ω, so x c limx→∞ () 1+=cx e. Theorem 3.6 tells us much more than this. In fact, it is now easy to show that x for any f ( x) satisfying * f (ω) < α , the limit limx→∞ () 1++cx f() x does not x differ from limx→∞ () 1+ cx . We can also use Theorem 3.6 to more easily evaluate limits of this form where c is a function of x instead of a constant. This is demonstrated in the first example below.

()1+ ln x x Example 3.7. lim x x→∞ x +1

Solution. We first evaluate at ω to obtain the expression

()1ln+ αωω . ω +1

Since lnω ∈ zone( 1), local equality on the level of α is equivalent to local equality on the level of α lnω . So 1ln+α ωω==α eαωln α, from which Theo-

28 rem 3.6 tells us that ()1ln+=α ω ω ω ω. We also have for the denominator ωω+=1 ω , so by Theorem 3.3,

()1ln+ αωω =1 1. ω +1

Therefore, the value of the limit is 1.

1 ⎛⎞abxx+ x Example 3.8. lim⎜⎟ x→0 ⎝⎠2

Solution. Evaluating at α, we have

ω ⎛⎞abαα+ ⎜⎟. ⎝⎠2

Using the identity ceαα= ln c, this can be rewritten as

ω ⎛⎞eeααlnab+ ln ⎜⎟. ⎝⎠2

Adding the power series for the exponentials together, we obtain

ω ⎛⎞112 22 ⎜⎟1lnlnlnln.+++++αα()ab()()() a b ⎝⎠24

Even though the above expression is not the power series for eab()lnab+ ln 2 = , it is locally equal to this power series on the level of α. So we can apply Theorem 3.6 to obtain

ω ⎛⎞112 22 ab ⎜⎟1lnlnlnln+++++=αα()ab()()() a b ab . ⎝⎠24

Therefore, the value of the limit is ab.

2 Example 3.9. lim() cos x cot x x→0

Solution. Evaluating at α and substituting power series, we have

29 2 ⎛⎞12!−+−α 2 2 ⎜⎟3 cot2 α ⎛⎞α ⎝⎠αα−+−3! ()cosα =−⎜⎟ 1 +− . ⎝⎠2!

2 For the exponent we can use Theorem 3.3 to obtain 12!−+−α =1 ω since αα−+−3 3! cosα =1 1 and sinα =1 α . So we write t = cot 22α =+ωβ where β < ω . Now 2 cot2 α for the base we have cosαα− 1== 1 t. All of this shows that ()cosα sa- −αα222 tisfies the conditions for Theorem 3.6. Since e = cosα and cosα = 1, Theorem 3.6 tells us that

2 2 2 cot α ()cosα cot α =12()e−α

2 2 ω +β = ()e−α 2

2 = ee−−12αβ 2. Since β < ω , −α 2β 2 is an infinitesimal which we will call γ. We now have

2 ee−−12αβ 2= ee − 12 γ ∞ n −12⎛⎞γ =+e ⎜⎟1.∑ ⎝⎠n=1 n!

Using Lemma 3.1,

∞∞γ nnγ ∑ ≤ ∑ nn==11nn!! ∞ < ∑ γ n n=1 = γ .

So we finally have

∞ n −−12⎛⎞γ 1 12 ee⎜⎟1.+=∑ ⎝⎠n=1 n!

Therefore, the value of the limit is e−12.

30 These examples would require a great deal more work if we were to use l’Hôpital’s rule to evaluate the limits. However, to one who is adept at using Theorems 3.3 and 3.6, these limits can be evaluated with far less effort.

3.4 Uncompensated Completion Another interesting situation arises for some limits which assume the form ∞−∞. The example following the next theorem shows that we may sometimes add a number to an expression in order to complete a square without ever sub- tracting the number elsewhere—that is, we never have to compensate for the change to the expression. Theorem 3.10. Let t be a positive infinite number and let ct< . Then tc+=1 t.

Proof. Expanding tc+ with the binomial theorem we have

∞ ⎛⎞12 tc+= t12−kk c ∑⎜⎟ k =0 ⎝⎠k ∞ ⎛⎞12 ck =+tt⎜⎟. (3.17) ∑⎜⎟k k =1 ⎝⎠k t

Since 12 <1 for all k, we can write ( k ) ∞∞⎛⎞12 cckk ⎜⎟≤ . (3.18) ∑⎜⎟kk∑ kk==11⎝⎠k tt Since ct< , ct is an infinitesimal. So Lemma 3.1 applies to this sum giving ∞ kk us ct= ct. We now have ∑ k =1 ( )

∞ ⎛⎞12 cck t ⎜⎟≤ . (3.19) ∑⎜⎟k k =1 ⎝⎠k t t Since ct< , ct<1. Therefore, from equation (3.17), tc+= t +ξ where ξ <1. So tc+=1 t.

Example 3.11. limx2 + 6xx− x→∞

31 Solution. Evaluating at ω, we have ω 2 + 6ω −ω. Theorem 3.10 tells us that for any c <+ω2 6ω , we must have ω 212++=66ωωωc +. So we choose the only value of c that is of any advantage—the one which com- pletes the square under the radical. Setting c = 9, we have

ω 212+ 669ωω−= ω + ω +− ω =+−()ωω3 2 = 3.

Therefore, the value of the limit is 3.

This method of uncompensated square completion provides a much faster alter- native to the standard method of multiplying the expression by its conjugate.

32 4 Infinite Series

n this section we present two new tests for the convergence of infinite series which are analogs of the comparison test and the limit comparison test of I standard analysis. Once these tests have been introduced, we examine some useful facts which allow easier application of the tests and present some exam- ples.

4.1 Convergence Tests Both of the new tests determine the convergence of a series ∞ f n by ex- ∑ n=1 () amining the properties of * f (ω). The first test checks to see whether ∞ **gfω < ω for some convergent series f n . ( ) ( ) ∑ n=1 () Theorem 4.1 (Order Comparison Test). Let f ( x) and g ( x) be standard positive-valued functions. If the series ∞ f ()n converges and ∑ n=1 ∞ **gfω < ω (independent on the choice of ω) then the series g n ( ) ( ) ∑ n=1 () also converges. Proof. Since **gf(ω) < (ω), we know that **gf(ω) (ω) < 1 and thus **gf(ω) (ω) is an infinitesimal. Since this happens independently of the

choice of ω, this implies that limx→∞ gx( ) f( x) = 0. Therefore, given any ε > 0, there exists an N ∈ N such that for all nN> , gn( ) f( n) < ε . Thus for ∞ ∞ all nN> , g nfn< ε . Since ε f n converges, g n also con- ( ) ( ) ∑ n=1 () ∑ n=1 () verges by the comparison test.

Note that the contrapositive of this theorem states that if the series ∞ g ()n ∞ ∑ n=1 diverges and **gfω < ω , then the series f n also diverges. ( ) ( ) ∑ n=1 () Let zg= ord( * (ω)) for some standard positive-valued function g. It imme- diately follows from the order comparison test that if z −1 then the series ∞ g ()n converges since in this case we would have *1g (ω) < ω p for some ∑ n=1 ∞ real number p with 1<<−p z. If z −1, then the series g ()n diverges since ∑ n=1 in this case we would have *1g (ω) > ω p for some real p with −

33 converges tells us nothing about whether ∞ g n converges. However, if ∑ n=1 () **fg()ω =* f ()ω ()ω then we can infer the convergence of one series from the other. For this situation, we have the following test.

Theorem 4.2 (Local Equality Test). If f and g are standard positive-valued functions that satisfy **fg(ω) =* f (ω) (ω) (independent on the choice of ω), then the series ∞ f n and ∞ g n either both converge or both di- ∑ n=1 () ∑ n=1 () verge.

Proof. Since **fg(ω) =* f (ω) (ω), we know that **gf(ωω)()=1 1, which implies that **gf(ω) (ω) 1. Since this property is independent of the

choice of ω, this means that limx→∞ gx( ) f( x) = 1. Therefore, by the limit comparison test, the series ∞ f n and ∞ g n either both converge or ∑ n=1 () ∑ n=1 () both diverge.

4.2 Using the New Tests It is usually more convenient to think of an infinite series as a sum of the reci- procals of a function evaluated at each natural number. Fortunately, the order comparison test and local equality test work equally well for this situation. This is because **fg(ω) > (ω) implies 1*fg(ω) < 1* (ω) and **fg()ω =* f ()ω (ω) implies 1*fg()ω =1*f (ω) 1* (ω). In many cases when the order comparison test cannot be used, it will be poss- ible through local equality to reduce the number * f (ω) to a number of the form 1 aω where a ∈ zone( 1). When this happens, the following extension to the p-series test is useful. We use the notation ln(n) to mean the tak- en n (e.g., ln()3 x = ln ln ln x). Let k ∈ N and let m be the least natural number for which ln()k m is real and greater than 1. Then the series

∞ 1

∑ ()2 ()k p nm= nnln ln n() ln n converges if and only if p > 1. We show this by using the test. The integral

34 ∞ 1 dx ∫m ()2 ()k p xxln ln x() ln x can be evaluated by making the substitution ux= ln(k ) for which we have du=1lnln x x()21 x ln(k− ) x. This gives us the integral

∞ du , ∫ln()k m u p which converges if and only if p > 1. We conclude this section with a few examples.

∞ nn2 − 32+ Example 4.3. ∑ 32 n=1 461nn+ −

Solution. Let fn= 461nn32+ − . We substitute ω for n and notice () nn2 −32+

461ωω32+− = ω. ωω2 −+32

So if we can find a function g n such that 1*g ω =ω 461ωω32+− for which ( ) () ωω2 −+32 we know whether ∞ 1 g n converges, then we can use the local equality ∑ n=1 () test to determine whether ∞ 1 f n converges. By Theorem 3.3, ∑ n=1 () 461ωω32+− =ω 4ω ωω2 −+32

3 2 since 4614ω32+−=ωωω 3 and ω 22−+=32ωωω . Because ∞ 14n di- ∑ n=1 verges, the local equality test tells us that the series that we are testing also diverges.

The next example demonstrates the procedure for dealing with factorials. As shown, the Stirling for the factorial gives a good representation of the size of ω!.

∞ n5 Example 4.4. ∑ n=1 n!

Solution. From Stirling's approximation to n!, we know that

35 2π nnnn e− lim= 1. n→∞ n!

Therefore ω!2 πω ωωe−ω. Using this, we can write

ωω55 = ω+ 1 ω! ω 2 e−ω 1 = . ωα()1−−9 1 ω 2lnω

9 1 Since 1−−∈2lnα ω zone( 1), the exponent of ω in this last expression is still ωα(1−−9 1 ) 2 an infinite number, so ω 2lnω > ω . Therefore, by the order comparison test, the series converges.

The above example may seem like a lot of work for such a simple summand. The intent was to demonstrate that ω !> ω p for any finite number p. This fact will usually be enough to tell quickly whether a series containing factorials con- verges. We finish with a short example.

∞ 1 Example 4.5. ∑ 2 n=1 nn()ln− ln n

Solution. The local equality test tells us that this series converges if and only if the series ∞ 1lnnn2 converges, which it does by the earlier remark ∑ n=1 () pertaining to the extended p-series test.

36 Bibliography

[1] Cutland, Nigel, ed., Nonstandard Analysis and its Applications, Cam- bridge University Press, 1988.

[2] Davis, Martin, Applied Nonstandard Analysis, Wiley-Interscience, 1977.

[3] Hurd, A. E. and Loeb, P. A., An Introduction to Nonstandard Real Anal- ysis, Academic Press, 1985.

[4] Keisler, H. Jerome, Elementary Calculus, Prindle, Weber, and Schmidt, 1976.

[5] Keisler, H. Jerome, Foundations of Infinitesimal Calculus, Prindle, We- ber, and Schmidt, 1976.

[6] Kusraev, Anatoly G. and Kutateladze, Semen S., Nonstandard Methods of Analysis, Kluwer Academic Publishers, 1994.

[7] Robert, Alain, Nonstandard Analysis, Wiley-Interscience, 1988.

[8] Robinson, Abraham, Non-standard Analysis, Princeton University Press, 1996.

[9] Robinson, Abraham and Lightstone, A. H., Nonarchimedean Fields and Asymptotic Expansions, North-Holland Publishing Company, 1975.

37 List of Notation

xnn≡ y xnn= y almost everywhere

[ an ] { xnn| xa≡ n}

*A *-transform of A; {[ aaAnn]|almost∈ everywhere} *N Hypernatural numbers *Z Hyperintegers *Q Hyperrational numbers *R Hyperreal numbers T Set of all infinite numbers S Set of all infinitesimals

A∞ Set of infinite numbers in A; A ∩ T x y x is infinitely close to y; x − y ∈S x y x < y or x y x y x > y or x y xy x < y and x / y xy x > y and x / y ma( ) Monad about a; {x | xa } Ga( ) Galaxy about a; {xx|− a is finite} ord(a) Order of a; logω a T′ {xx∈∉TS|ord( ) } S′ {xx∈∉SS|ord( ) } x = y x is isometric to y; ord( x) ord( y) x < y x is inferior to y; ord( xy) ord( ) x > y x is superior to y; ord( xy) ord( ) xy≤ x < y or x = y xy≥ x > y or x = y zone a Zone about a; x | xa= ( ) { } Wa( ) World about a; {x | xa≤ } Wa( ) {x | xa< } 0 xy= x is locally equal to y on the level of ; x − y <

38