sophisticated limits with focus on mathematical olympiad problems By tetyana darian A thesis submitted to the Graduate School-Camden Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Master of Science Graduate Program in Pure Mathematics Written under the direction of Gabor Toth And approved by Gabor Toth Camden, New Jersey October 2019 thesis abstract Sophisticated Limits with Focus on Mathematical Olympiad Problems by tetyana darian Thesis Director: Gabor Toth The purpose of this thesis is to derive a variety of sophisticated limits, some in Mathematical Olympiad Problems, without using mathematically advanced concepts. In fact, this thesis requires only basic calculus and the Stolz-Ces`arotheorems. The latter will be presented without proofs but with all the necessary ingredients and formulas needed in this work. The thesis is written for a mathematically mature student with a certain level of preparation to tackle these beautiful limits. ii Acknowledgements I would like to thank my thesis advisor, Professor Gabor Toth, for his thoughtful guidance, support, and intuition. I truly appreciate and have enjoyed all of our meetings, discussions, and classroom time together. Thank you for introducing me to the beautiful and complex world of mathematics and mathematical research. A separate thank you and appreciation goes to my husband, Steven, who supported me through thick and thin during my mathematical journey. iii Contents Thesis Abstract ii Acknowledgements iii 1 Introduction 1 1.1 Completeness of the Real Numbers . 1 1.2 Neighborhoods and Limit Points . 2 1.3 The Limit of a Sequence . 2 2 Stolz-Ces`aroTheorems and the Stirling Formula 4 2.1 The Stolz-Ces`aroTheorem . 4 2.2 The Stolz-Ces`aroTheorem . 5 2.3 The Stirling Formula . 7 3 Examples of Sophisticated Limits 8 3.1 Preliminaries . 8 3.2 Example 1 . 9 3.3 Example 2 . 13 3.4 Example 3 . 17 3.5 Example 4 . 20 3.6 Example 5 . 21 3.7 Example 6 . 23 Bibliography 25 iv 1 Chapter 1 Introduction This chapter is a brief look at what this thesis is all about. To aid us in our discussion, we will be including some definitions and results which will be stated without proof. Important: All definitions, propositions, and theorems presented in this chapter will be reintroduced, and elaborated upon throughout the course of the thesis. The results of this chapter will be used without any explicit references. To further emphasize this, we will not number the definitions, propositions, and theorems of this chapter. 1.1 Completeness of the Real Numbers Definition. A nonempty set S of real numbers is said to be bounded above if there exists some real number M such that x ≤ M for every x in S. In this case M is said to be an upper bound for S. Likewise, S is said to be bounded below if there exists some number m such that m ≤ x for all x in S; m is called a lower bound for S. We call S bounded if it is bounded both above and below. Definition. Let S be a nonempty set of real numbers that is bounded above. A supremum or least upper bound of S, denoted sup S, is a real number µ such that 2 1. x ≤ µ, for all x in S; 2. If M is an upper bound for S, then µ ≤ M. Definition. Let S be a nonempty set of real numbers that is bounded below. An infimum or greatest lower bound of S, denoted inf S, is a real number ν such that 1. ν ≤ x, for all x in S; 2. If m is any lower bound for S, then ν ≥ m. Axiom (The Completeness Axiom for R). If S is a nonempty set of real numbers that is bounded above, then sup S exists in R. 1.2 Neighborhoods and Limit Points Definition. Let x be any real number. The neighborhood of x with positive radius r is the set N(x; r) = fy in R : jy − xj < rg Definition. A deleted neighborhood of x in R, denoted N 0(x; r), is the neighborhood N(x; r) with the point x itself removed. Definition. Given a nonempty set S in R, a point x in R is said to be a limit point of S if, for each " > 0, the deleted neighborhood N(x; ") contains at least one point of S. Theorem (Bolzano-Weierstrass Theorem). If S is a bounded, infinite subset of R, then S has a limit point in R. 1.3 The Limit of a Sequence Definition. The point c is a cluster point of the sequence fxkg if, for every " > 0 and every k in N, there is a k1 > k such that xk1 is in N(c; "). 3 Definition. The sequence fxkg converges to x0 and we say x0 is the limit of fxkg if, for each neighborhood N(x0; "), there exists an index k0 such that, whenever k > k0, xk is in N(x0; "). We write limk!1 xk = x0. If a sequence fxkg fails to converge, for whatever reason, then we say that the sequence diverges. Theorem. The limit of a convergent sequence in R is unique. Theorem. A convergent sequence is bounded. Theorem. If a sequence is unbounded, then it must diverge. Theorem. A bounded, monotonic sequence of real numbers converges. Definition. Let fxkg be any sequence. Choose any strictly monotonic increasing sequence k1 < k2 < k3 < ::: of natural numbers. For each j in N, let yj = xkj . The sequence fyjg = fxkj g is called a subsequence of fxkg. Theorem. The point c is a cluster point of fxkg if and only if there exists a subsequence xkj that converges to c. Theorem. Any bounded sequence has a cluster point. Theorem. A sequence xkj converges to x0 if and only if every subsequence of xkj converges to x0. Theorem. A bounded sequence converges if and only if it has exactly one cluster point. 4 Chapter 2 Stolz-Ces`aroTheorems and the Stirling Formula In this chapter, we discuss powerful criteria for convergence of sequences, all due to Otto Stolz and Ernesto Ces`aro. 2.1 The Stolz-Ces`aroTheorem Let (a ) and (b ) be real sequences such that (b ) is strictly increasing n n2N0 n n2N0 n n2N0 with limn!1 bn = 1. Then, we have a − a a a a − a lim inf n n−1 ≤ lim inf n ≤ lim sup n ≤ lim sup n n−1 : n!1 bn − bn−1 n!1 bn n!1 bn n!1 bn − bn−1 In particular a − a a lim n n−1 = lim n ; n!1 bn − bn−1 n!1 bn provided that the limit on the left-hand side exists. 5 2.2 The Stolz-Ces`aroTheorem (Equivalent Formulation) Let (a ) and (b ) be real sequences such that b > 0; n 2 and n n2N0 n n2N0 n N0 limn!1 bn = 1. Then, we have a a + ··· + a a + ··· + a a lim inf n ≤ lim inf 1 n ≤ lim sup 1 n ≤ lim sup n : n!1 bn n!1 b1 + ··· + bn n!1 b1 + ··· + bn n!1 bn In particular a a + ··· + a lim n = lim 1 n ; n!1 bn n!1 b1 + ··· + bn provided that the limit on the left-hand side exists. Proof. This follows directly from the previous by the substitution an 7! a0 + ··· + an and bn 7! b0 + ··· + bn, n 2 N0. Letting bn = n; n 2 N, we obtain the following special cases valid for any real sequence (a ) : n n2N a1 + ··· + an a1 + ··· + an lim inf an ≤ lim inf ≤ lim sup ≤ lim sup an; n!1 n!1 n n!1 n n!1 and an an lim inf (an − an−1) ≤ lim inf ≤ lim sup ≤ lim sup (an − an−1) : n!1 n!1 n n!1 n n!1 In particular, a1 + ··· + an lim an = lim ; n!1 n!1 n and an lim (an − an−1) = lim ; n!1 n!1 n provided that the limits on the left-hand sides exist. 6 We call these the additive Stolz-Ces`aro formulas. Let (a ) be a real sequence with positive members. For n 2 , let b = log (a ), n n2N N n 2 n bn or equivalently, an = 2 . Applying the exponential identities, we obtain b1+···+bn p n 2 n = a1 : : : an: Using the monotonicity of the exponentiation, and the Stolz-Ces`arolimit formulas above for the sequence (b ) , we obtain n n2N p p n n lim inf an ≤ lim inf a1 : : : an ≤ lim sup a1 : : : an ≤ lim sup an; n!1 n!1 n!1 n!1 and a p p a n n n n lim inf ≤ lim inf an ≤ lim sup an ≤ lim inf : n!1 an−1 n!1 n!1 n!1 an−1 In particular p n lim an = lim a1 : : : an; n!1 n!1 and a p n n lim = lim an; an−1 n!1 provided that the limits on the left-hand sides exist. We call these the multiplicative Stolz-Ces`aro formulas. 7 2.3 The Stirling Formula An alternative approach for deriving some of the limits that follow in Chapter 3 is to first derive the Stirling Formula. However, the Stirling Formula is much more involved, compared to the more elementary Stolz-Ces`aroformulas, and would require elaborate proofs that are not within the scope of this work. 8 Chapter 3 Examples of Sophisticated Limits 3.1 Preliminaries We begin with a preliminary example: n (1) lim p = e: n!1 n n! n Indeed, letting an = n!=n , n 2 N, we calculate n n an+1 (n + 1)! n n 1 1 lim = lim n+1 · = lim n = lim n = ; n!1 an n!1 (n + 1) n! n!1 (n + 1) n!1 (1 + 1=n) e where we use the Euler's limit relation.