DOCSLIB.ORG

1.4 Cauchy Sequence in R

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1.4 Cauchy Sequence in R 1.4 Cauchy Sequence in R De¯nition. (1.4.1) A sequence xn 2 R is said to converge to a limit x if ² 8² > 0; 9N s:t: n > N ) jxn ¡ xj < ²: A sequence xn 2 R is called Cauchy sequence if ² 8²; 9N s:t: n > N & m > N ) jxn ¡ xmj < ²: Proposition. (1.4.2) Every convergent sequence is a Cauchy sequence. Proof. Assume xk ! x. Let ² > 0 be given. ² ² 9N s:t: n > N ) jxn ¡ xj < 2 . ² n; m ¸ N ) ² ² jxn ¡ xmj · jx ¡ xnj + jx ¡ xmj < 2 + 2 = ²: 1 Theorem. (1.4.3; Bolzano-Weierstrass Property) Every bounded sequence in R has a subsequence that converges to some point in R. Proof. Suppose xn is a bounded sequence in R. 9M such that ¡M · xn · M; n = 1; 2; ¢ ¢ ¢ . Select xn0 = x1. ² Bisect I0 := [¡M; M] into [¡M; 0] and [0; M]. ² At least one of these (either [¡M; 0] or [0; M]) must contain xn for in¯nitely many indices n. ² Call it I1 and select n1 > n0 with xn1 2 I0. ² Continue in this way to get a subsequence xnk such that ² I0 ⊃ I1 ⊃ I2 ⊃ I3 ¢ ¢ ¢ ¡k ² Ik = [ak ; bk ] with jIk j = 2 M. ² Choose n0 < n1 < n2 < ¢ ¢ ¢ with xnk 2 Ik . 9 ² Since ak · ak+1 · M (monotone and bounded), ak ! x. ¡k ² Since xnk 2 Ik and jIk j = 2 M, we have ¡k¡1 jxnk ¡xj < jxnk ¡ak j+jak ¡ xj · 2 M+jak ¡ xj ! 0 as k ! 1: 2 Corollary. (1.4.5; Compactness) Every sequence in the closed interval [a; b] has a subsequence in R that converges to some point in R. Proof. Assume a · xn · b for n = 1; 2; ¢ ¢ ¢ . By Theorem 1.4.3, 9 9 a subsequence xnk and a · x · b such that xnk ! x. Lemma. (1.4.6; Boundedness of Cauchy sequence) If xn is a Cauchy sequence, xn is bounded. Proof. 9N s.t. n ¸ N ) jxn ¡ xj < 1. Then supn jxnj · 1 + maxfjx1j; ¢ ¢ ¢ ; jxN jg (Why?) Theorem. (1.4.3; Completeness) Every Cauchy sequence in R converges to an element in [a; b]. Proof. Cauchy seq. ) bounded seq. ) convergent subseq. 3 1.5. Cluster Points of the sequence xn De¯nition. (1.5.1; cluster points) A point x is called a cluster point of the sequence xn if ² 8² > 0; 9in¯nitely many values of n with jxn ¡ xj < ² In other words, a point x is a cluster point of the sequence xn i® 8² > 0 & 8N; 9n > N s:t: jxn ¡ xj < ² Example ² Both 1 and ¡1 are cluster points of the sequence 1; ¡1; 1; ¡1; ¢ ¢ ¢ : 1 ² The sequence xn = n has the only cluster point 0. ² The sequence xn = n does not have any cluster point. 4 Proposition. 1. x is a cluster point of the sequence xn i® 9 a subsequence xnk s:t: xnk ! x: 2. xn ! x i® every subsequence of xn converges to x 3. xn ! x i® the sequence fxng is bounded and x is its only cluster points. Proof. 1. ()) Assume x is a cluster point. Then, we can choose 1 n1 < n2 < n3 ¢ ¢ ¢ s:t: jxnk ¡ xj < k . (Why?) This gives a subsequence xnk ! x. 2. Trivial 3. (()If not, 9² and 9 a subseq xnk so that jxnk ¡ xj > ². Since xnk is bounded, 9 a convergent subseq. The limit of that subseq would be a cluster pt of the seq xn di®erent from x, but there are no such pt. Contradiction. 5 De¯nition. (1.5.3; limit superior & limit inferior of seq xn ) De¯ne the limit superior limxn in the following way: ² If xn is bounded above, then lim supn!1 xn = limxn = the largest cluster point limxn = ¡1 if the set cluster point is empty ² If xn is NOT bounded above, then limxn = 1 Similarly, we can de¯ne the limit inferior limxn. Examples ² For the seq 1; 0; ¡1; 1; 0; ¡1; ¢ ¢ ¢ , limxn = 1 and limxn = ¡1. ² If xn = n, then limxn = 1 = limxn n 1+n ² Let xn = (¡1) n . Then limxn = 1 and limxn = ¡1. 6 De¯nition. (1.6.2; Vector space) A real vector space V is a set of elements called vectors, with given operations of vector addition + : V £ V ! V and scalar multiplication ¢ : R £ V ! V such that the followings hold for all v; u; w 2 V and all ¸; ¹ 2 R: 1. v + w = w + v; (v + u) + w = v + (u + w); ¸(v + w) = ¸v + ¸w; ¸(¹v) = (¸¹)v; (¸ + ¹)v = ¸v + ¹v; 1v = v. 2. 90 2 V s.t. v + 0 = v. 9 ¡ v 2 V s.t. v ¡ v = 0. ² A subset of V is called a subspace if it is itself a vector space with the same operations. ² W is a vector subspace of V i® ¸v + ¹u 2 W whenever u; v 2 W and ¸; ¹ 2 R. ² The straight line W = f(x1; x2): x1 = 2x2g is a subspace of R2. 7 Euclidean space Rn & De¯nitions & Properties The Euclidean n-space Rn with the operations (x1; ¢ ¢ ¢ ; xn) + (y1; ¢ ¢ ¢ ; yn) = (x1 + y1; ¢ ¢ ¢ ; xn + yn)& ¸(x1; ¢ ¢ ¢ ; xn) = (¸x1; ¢ ¢ ¢ ; ¸xn) is a vector space of dimension n. ² The standard basis of Rn; e1 = (1; 0; ¢ ¢ ¢ ; 0); ¢ ¢ ¢ ; en = (0; ¢ ¢ ¢ ; 0; 1). n ² Unique representation: x = (x1; ¢ ¢ ¢ ; xn) 2 R can be expressed uniquely as x = x1e1 + ¢ ¢ ¢ + xnen: P ² Inner product of x and y: hx; yi = n x y p i=1 i i ² Norm of x: kxk = hx; yi. ² Distance between x and y: dist(x; y) = kx ¡ yk ² Triangle inequality: kx + yk · kxk + kyk. ² Cauchy-Schwartz inequality: hx; yi · kxk kyk ² Pythagorean theorem: If hx; yi = 0, then kx + yk2 = kxk2 + kyk2. 8 De¯nition. (1.7.1; Metric Space (M; d) equipped with d =distance) A metric space (M; d) is a set M and a function d : M £ M ! R such that 1. d(x; y) ¸ 0 for all x; y 2 M. 2. d(x; y) = 0 i® x = y. 3. d(x; y) = d(y; x) for all x; y 2 M. 4. d(x; y) · d(x; z) + d(z; y) for all x; y 2 M. Example [Fingerprint Recognition] Let M be a data set of ¯ngerprints in Seoul city police department. ² Motivation: Design an e±cient access system to ¯nd a target. ² We need to de¯ne a dissimilarity function stating the distance between the data. The distance d(x; y) between two data x and y must satisfy the above four rules. ² Similarity queries. For a given target x¤ 2 M and ² > 0, arrest all having ¯nger print y 2 M such that d(y; x¤) < ². 9 De¯nition. (1.7.3. Normed Space (V; k ¢ k)) A normed space (V; k ¢ k) is a vector space V and a function k ¢ k : V! R called a norm such that 1. kvk ¸ 0; 8v 2 V 2. kvk = 0 i® v = 0. 3. k¸vk = j¸jkvk; 8v 2 V and every scaler ¸. 4. kv + wk · kvk + kwk; 8v; w 2 V Examples ² V = R and kxk = jxj for all x 2 R. q 2 2 2 2 ² V = R and kvk = v1 + v2 for all v = (v1; v2) 2 R . ² Let V = C([0; 1])=all continuous functions on the interval [a; b]. De¯ne kf k = supfjf (x)j : x 2 [0; 1]g (called supremum norm): 10 Proposition. If (V; k ¢ k) is a normed vector space and d(v; w) = kv ¡ wk , then d is a metric in V. Proof. EASY. Examples ² For V = C([0; 1]), the metric is d(f ; g) = kf ¡ gk = supfjf (x) ¡ g(x)j : x 2 [0; 1]g: The sup distance between functions is the largest vertical distance between their graphs. 11 De¯nition. A vector space V with a function h¢; ¢i : V £ V ! R is called an inner product space if 1. h v; vi ¸ 0 for all v 2 V. 2. h v; vi = 0 i® v = 0. 3. h¸v; wi = ¸hv; wi; 8v 2 V and every scaler ¸. 4. hv + w; hi = h v; h i + h w; h i. 5. h v; w i = h w; v i Examples 2 1. V = R and h v; w i = v1w1 + v2w2. Two vectors v and w are orthogonal if h v; w i = 0. R 2. V = C[0; 1] and h f ; g i = 1 f (x)g(x)dx p 0 3. kvk = h v; v i is a norm on V. 12 Theorem. (Cauchy-Schwarz inequality ) If h ¢; ¢ i is an inner product in a real vector space V, then jh f ; g ij · kf kkgk Proof: ² g Suppose g 6= 0. Let h = kgk . It su±ces to prove that jh f ; h ij · kf k. (Why? jh f ; g ij · kf kkgk i® jh f ; h ij · kf k.) ² Denote ® = h f ; h i. Then 0 · kf ¡ ®hk2 = hf ¡ ®h; f ¡ ®hi = kf k2 ¡ ® hh; f i ¡ ® hf ; hi + j®j2 = kf k2 ¡ j®j2 Hence, j®j = jh f ; h ij · kf k. This completes the proof. 13 Chapter 2: Topology of M = Rn n Throughout this chapter, assumepP M = R ( the Euclidean space ) n 2 with the metric d(x; y) = i=1 jxi ¡ yi j = kx ¡ yk De¯nition. (D(x; ²), open, neighborhood) ² D(x; ²) := fy 2 M : d(y; x) < ²g is called ²-ball (or ²-disk) about x. ² A ½ M is open if 8 x 2 M; 9² > 0 s.t. D(x; ²) ½ A. ² A neighborhood of x is an open set A containing x.
Load more

Recommended publications
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • Sequence Rules
    SEQUENCE RULES A connected series of five of the same colored chip either up or THE JACKS down, across or diagonally on the playing surface. There are 8 Jacks in the card deck. The 4 Jacks with TWO EYES are wild. To play a two-eyed Jack, place it on your discard pile and place NOTE: There are printed chips in the four corners of the game board. one of your marker chips on any open space on the game board. The All players must use them as though their color marker chip is in 4 jacks with ONE EYE are anti-wild. To play a one-eyed Jack, place the corner. When using a corner, only four of your marker chips are it on your discard pile and remove one marker chip from the game needed to complete a Sequence. More than one player may use the board belonging to your opponent. That completes your turn. You same corner as part of a Sequence. cannot place one of your marker chips on that same space during this turn. You cannot remove a marker chip that is already part of a OBJECT OF THE GAME: completed SEQUENCE. Once a SEQUENCE is achieved by a player For 2 players or 2 teams: One player or team must score TWO SE- or a team, it cannot be broken. You may play either one of the Jacks QUENCES before their opponents. whenever they work best for your strategy, during your turn. For 3 players or 3 teams: One player or team must score ONE SE- DEAD CARD QUENCE before their opponents.
    [Show full text]
  • 1.1 Constructing the Real Numbers
    18.095 Lecture Series in Mathematics IAP 2015 Lecture #1 01/05/2015 What is number theory? The study of numbers of course! But what is a number? • N = f0; 1; 2; 3;:::g can be defined in several ways: { Finite ordinals (0 := fg, n + 1 := n [ fng). { Finite cardinals (isomorphism classes of finite sets). { Strings over a unary alphabet (\", \1", \11", \111", . ). N is a commutative semiring: addition and multiplication satisfy the usual commu- tative/associative/distributive properties with identities 0 and 1 (and 0 annihilates). Totally ordered (as ordinals/cardinals), making it a (positive) ordered semiring. • Z = {±n : n 2 Ng.A commutative ring (commutative semiring, additive inverses). Contains Z>0 = N − f0g closed under +; × with Z = −Z>0 t f0g t Z>0. This makes Z an ordered ring (in fact, an ordered domain). • Q = fa=b : a; 2 Z; b 6= 0g= ∼, where a=b ∼ c=d if ad = bc. A field (commutative ring, multiplicative inverses, 0 6= 1) containing Z = fn=1g. Contains Q>0 = fa=b : a; b 2 Z>0g closed under +; × with Q = −Q>0 t f0g t Q>0. This makes Q an ordered field. • R is the completion of Q, making it a complete ordered field. Each of the algebraic structures N; Z; Q; R is canonical in the following sense: every non-trivial algebraic structure of the same type (ordered semiring, ordered ring, ordered field, complete ordered field) contains a copy of N; Z; Q; R inside it. 1.1 Constructing the real numbers What do we mean by the completion of Q? There are two possibilities: 1.
    [Show full text]
  • 0.999… = 1 an Infinitesimal Explanation Bryan Dawson
    0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder .
    [Show full text]
  • Be a Metric Space
    2 The University of Sydney show that Z is closed in R. The complement of Z in R is the union of all the Pure Mathematics 3901 open intervals (n, n + 1), where n runs through all of Z, and this is open since every union of open sets is open. So Z is closed. Metric Spaces 2000 Alternatively, let (an) be a Cauchy sequence in Z. Choose an integer N such that d(xn, xm) < 1 for all n ≥ N. Put x = xN . Then for all n ≥ N we have Tutorial 5 |xn − x| = d(xn, xN ) < 1. But xn, x ∈ Z, and since two distinct integers always differ by at least 1 it follows that xn = x. This holds for all n > N. 1. Let X = (X, d) be a metric space. Let (xn) and (yn) be two sequences in X So xn → x as n → ∞ (since for all ε > 0 we have 0 = d(xn, x) < ε for all such that (yn) is a Cauchy sequence and d(xn, yn) → 0 as n → ∞. Prove that n > N). (i)(xn) is a Cauchy sequence in X, and 4. (i) Show that if D is a metric on the set X and f: Y → X is an injective (ii)(xn) converges to a limit x if and only if (yn) also converges to x. function then the formula d(a, b) = D(f(a), f(b)) defines a metric d on Y , and use this to show that d(m, n) = |m−1 − n−1| defines a metric Solution.
    [Show full text]
  • Arithmetic and Geometric Progressions
    Arithmetic and geometric progressions mcTY-apgp-2009-1 This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known as arithmetic progressions (APs) and geometric progressions (GPs), and the corresponding series. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • recognise the difference between a sequence and a series; • recognise an arithmetic progression; • find the n-th term of an arithmetic progression; • find the sum of an arithmetic series; • recognise a geometric progression; • find the n-th term of a geometric progression; • find the sum of a geometric series; • find the sum to infinity of a geometric series with common ratio r < 1. | | Contents 1. Sequences 2 2. Series 3 3. Arithmetic progressions 4 4. The sum of an arithmetic series 5 5. Geometric progressions 8 6. The sum of a geometric series 9 7. Convergence of geometric series 12 www.mathcentre.ac.uk 1 c mathcentre 2009 1. Sequences What is a sequence? It is a set of numbers which are written in some particular order. For example, take the numbers 1, 3, 5, 7, 9, .... Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained by adding 2 to give the next odd number.
    [Show full text]
  • 3 Formal Power Series
    MT5821 Advanced Combinatorics 3 Formal power series Generating functions are the most powerful tool available to combinatorial enu- merators. This week we are going to look at some of the things they can do. 3.1 Commutative rings with identity In studying formal power series, we need to specify what kind of coefficients we should allow. We will see that we need to be able to add, subtract and multiply coefficients; we need to have zero and one among our coefficients. Usually the integers, or the rational numbers, will work fine. But there are advantages to a more general approach. A favourite object of some group theorists, the so-called Nottingham group, is defined by power series over a finite field. A commutative ring with identity is an algebraic structure in which addition, subtraction, and multiplication are possible, and there are elements called 0 and 1, with the following familiar properties: • addition and multiplication are commutative and associative; • the distributive law holds, so we can expand brackets; • adding 0, or multiplying by 1, don’t change anything; • subtraction is the inverse of addition; • 0 6= 1. Examples incude the integers Z (this is in many ways the prototype); any field (for example, the rationals Q, real numbers R, complex numbers C, or integers modulo a prime p, Fp. Let R be a commutative ring with identity. An element u 2 R is a unit if there exists v 2 R such that uv = 1. The units form an abelian group under the operation of multiplication. Note that 0 is not a unit (why?).
    [Show full text]
  • Formal Power Series License: CC BY-NC-SA
    Formal Power Series License: CC BY-NC-SA Emma Franz April 28, 2015 1 Introduction The set S[[x]] of formal power series in x over a set S is the set of functions from the nonnegative integers to S. However, the way that we represent elements of S[[x]] will be as an infinite series, and operations in S[[x]] will be closely linked to the addition and multiplication of finite-degree polynomials. This paper will introduce a bit of the structure of sets of formal power series and then transfer over to a discussion of generating functions in combinatorics. The most familiar conceptualization of formal power series will come from taking coefficients of a power series from some sequence. Let fang = a0; a1; a2;::: be a sequence of numbers. Then 2 the formal power series associated with fang is the series A(s) = a0 + a1s + a2s + :::, where s is a formal variable. That is, we are not treating A as a function that can be evaluated for some s0. In general, A(s0) is not defined, but we will define A(0) to be a0. 2 Algebraic Structure Let R be a ring. We define R[[s]] to be the set of formal power series in s over R. Then R[[s]] is itself a ring, with the definitions of multiplication and addition following closely from how we define these operations for polynomials. 2 2 Let A(s) = a0 + a1s + a2s + ::: and B(s) = b0 + b1s + b1s + ::: be elements of R[[s]]. Then 2 the sum A(s) + B(s) is defined to be C(s) = c0 + c1s + c2s + :::, where ci = ai + bi for all i ≥ 0.
    [Show full text]
  • Formal Power Series Rings, Inverse Limits, and I-Adic Completions of Rings
    Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely many variables over a ring R, and show that it is Noetherian when R is. But we begin with a definition in much greater generality. Let S be a commutative semigroup (which will have identity 1S = 1) written multi- plicatively. The semigroup ring of S with coefficients in R may be thought of as the free R-module with basis S, with multiplication defined by the rule h k X X 0 0 X X 0 ( risi)( rjsj) = ( rirj)s: i=1 j=1 s2S 0 sisj =s We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s 2 S, f(s1; s2) 2 S × S : s1s2 = sg is finite. Thus, each element of S has only finitely many factorizations as a product of two k1 kn elements. For example, we may take S to be the set of all monomials fx1 ··· xn : n (k1; : : : ; kn) 2 N g in n variables. For this chocie of S, the usual semigroup ring R[S] may be identified with the polynomial ring R[x1; : : : ; xn] in n indeterminates over R. We next construct a formal semigroup ring denoted R[[S]]: we may think of this ring formally as consisting of all functions from S to R, but we shall indicate elements of the P ring notationally as (possibly infinite) formal sums s2S rss, where the function corre- sponding to this formal sum maps s to rs for all s 2 S.
    [Show full text]
  • Absolute Convergence in Ordered Fields
    Absolute Convergence in Ordered Fields Kristine Hampton Abstract This paper reviews Absolute Convergence in Ordered Fields by Clark and Diepeveen [1]. Contents 1 Introduction 2 2 Definition of Terms 2 2.1 Field . 2 2.2 Ordered Field . 3 2.3 The Symbol .......................... 3 2.4 Sequential Completeness and Cauchy Sequences . 3 2.5 New Sequences . 4 3 Summary of Results 4 4 Proof of Main Theorem 5 4.1 Main Theorem . 5 4.2 Proof . 6 5 Further Applications 8 5.1 Infinite Series in Ordered Fields . 8 5.2 Implications for Convergence Tests . 9 5.3 Uniform Convergence in Ordered Fields . 9 5.4 Integral Convergence in Ordered Fields . 9 1 1 Introduction In Absolute Convergence in Ordered Fields [1], the authors attempt to dis- tinguish between convergence and absolute convergence in ordered fields. In particular, Archimedean and non-Archimedean fields (to be defined later) are examined. In each field, the possibilities for absolute convergence with and without convergence are considered. Ultimately, the paper [1] attempts to offer conditions on various fields that guarantee if a series is convergent or not in that field if it is absolutely convergent. The results end up exposing a reliance upon sequential completeness in a field for any statements on the behavior of convergence in relation to absolute convergence to be made, and vice versa. The paper makes a noted attempt to be readable for a variety of mathematic levels, explaining new topics and ideas that might be confusing along the way. To understand the paper, only a basic understanding of series and convergence in R is required, although having a basic understanding of ordered fields would be ideal.
    [Show full text]
  • A Mathematical Example: Factorial Example: Fibonnaci Sequence Fibonacci As a Recursive Function Fibonacci: # of Frames Vs. # Of
    10/14/15 A Mathematical Example: Factorial Example: Fibonnaci Sequence • Non-recursive definition: • Sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ... a0 a1 a2 a3 a4 a5 a6 n! = n × n-1 × … × 2 × 1 § Get the next number by adding previous two = n (n-1 × … × 2 × 1) § What is a8? • Recursive definition: • Recursive definition: § a = a + a Recursive Case n! = n (n-1)! for n ≥ 0 Recursive case n n-1 n-2 § a0 = 1 Base Case 0! = 1 Base case § a1 = 1 (another) Base Case What happens if there is no base case? Why did we need two base cases this time? Fibonacci as a Recursive Function Fibonacci: # of Frames vs. # of Calls def fibonacci(n): • Function that calls itself • Fibonacci is very inefficient. """Returns: Fibonacci no. an § Each call is new frame § fib(n) has a stack that is always ≤ n Precondition: n ≥ 0 an int""" § Frames require memory § But fib(n) makes a lot of redundant calls if n <= 1: § ∞ calls = ∞ memo ry return 1 fib(5) fibonacci 3 Path to end = fib(4) fib(3) return (fibonacci(n-1)+ n 5 the call stack fibonacci(n-2)) fib(3) fib(2) fib(2) fib(1) fibonacci 1 fibonacci 1 fib(2) fib(1) fib(1) fib(0) fib(1) fib(0) n 4 n 3 fib(1) fib(0) How to Think About Recursive Functions Understanding the String Example 1. Have a precise function specification. def num_es(s): • Break problem into parts """Returns: # of 'e's in s""" 2. Base case(s): # s is empty number of e’s in s = § When the parameter values are as small as possible if s == '': Base case number of e’s in s[0] § When the answer is determined with little calculation.
    [Show full text]
  • Infinitesimals Via Cauchy Sequences: Refining the Classical Equivalence
    INFINITESIMALS VIA CAUCHY SEQUENCES: REFINING THE CLASSICAL EQUIVALENCE EMANUELE BOTTAZZI AND MIKHAIL G. KATZ Abstract. A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number sys- tem. Such an approach can be seen as formalizing Cauchy’s senti- ment that a null sequence “becomes” an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor’s belittling of infinitesimals. June 2, 2021 1. Historical background Robinson developed his framework for analysis with infinitesimals in his 1966 book [1]. There have been various attempts either (1) to obtain clarity and uniformity in developing analysis with in- finitesimals by working in a version of set theory with a concept of infinitesimal built in syntactically (see e.g., [2], [3]), or (2) to define nonstandard universes in a simplified way by providing mathematical structures that would not have all the features of a Hewitt–Luxemburg-style ultrapower [4], [5], but nevertheless would suffice for basic application and possibly could be helpful in teaching. The second approach has been carried out, for example, by James Henle arXiv:2106.00229v1 [math.LO] 1 Jun 2021 in his non-nonstandard Analysis [6] (based on Schmieden–Laugwitz [7]; see also [8]), as well as by Terry Tao in his “cheap nonstandard analysis” [9]. These frameworks are based upon the identification of real sequences whenever they are eventually equal.1 A precursor of this approach is found in the work of Giuseppe Peano. In his 1910 paper [10] (see also [11]) he introduced the notion of the end (fine; pl.
    [Show full text]