16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
|
|
- Darren Miles
- 6 years ago
- Views:
Transcription
1 Infinite Series 6. Introduction We extend the concet of a finite series, met in section, to the situation in which the number of terms increase without bound. We define what is meant by an infinite series being convergent by considering the artial sums of the series. As rime examles of infinite series we examine the harmonic and the alternating harmonic series and show that the former is divergent and the latter is convergent. We consider various tests for the convergence of series, in articular we introduce the Ratio test which is a test alicable to series of ositive terms. Finally we define the meaning of the terms absolute and conditional convergence. Prerequisites Before starting this Section you should... Learning Outcomes After comleting this Section you should be able to... be able to use the summation notation be familiar with the roerties of limits 3 be able to use inequalities use the alternating series test and the ratio test on infinite series understand the terms absolute and conditional convergence
2 . Introduction Many of the series considered in section were examles of finite series in that they all involved the summation of a finite number of terms. When the number of terms in the series increases without bound we refer to the sum as an infinite series. Of articular concern with infinite series is whether they are convergent or divergent. For examle, the infinite series ++++ is clearly divergent because the sum of the first n terms increases without bound as more and more terms are taken. It is less clear as to whether the harmonic and alternating harmonic series: converge or diverge. Indeed you may be surrised to find that the first is divergent and the second is convergent. What we shall do in this section is to consider some simle convergence tests for infinite series. Although we all have an intuitive idea as to the meaning of convergence of an infinite series we must be more recise in our aroach. We need a definition for convergence which we can aly rigorously. First, using an obvious extension of the notation we have used for a finite sum of terms we denote the infinite series: a + a + a a + by the exression where a is an exression for the th term in the series. So, as examles: a ++3+ = = since the th term is a since the th term is a = ( ) + here a ( )+ Consider the infinite series: a + a + + a + = a What we do is to consider the sequence of artial sums, S,S,...,ofthis series where S = a S = a + a. S n = a + a + + a n HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
3 That is, S n is the sum of the first n terms of the infinite series. If the limit of the sequence S,S,...,S n,... can be found; that is lim S n = S n then we define the sum of the infinite series to be S: S = a (say) and we say the series converges to S. Another way of stating this is to say that a = lim n n a Definition An infinite series Convergence of Infinite Series a is convergent if the sequence of artial sums S,S,S 3,...,S k,... in which S k = k a is convergent Divergence condition for an infinite series An almost obvious requirement that an infinite series should be convergent is that the individual terms in the series should get smaller and smaller. This leads to the following keyoint: The condition: Key Point a 0 as increases (mathematically lim a =0) is a necessary condition for the convergence of the series a It is not ossible for an infinite series to be convergent unless this condition holds. 3 HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
4 Which of the following series cannot be convergent? (a) (b) (c) In each case, use the condition from the revious Keyoint. (a) a = lim a = a = lim + = + Hence series is divergent. (b) a = lim a = a = lim a =0 so this series may be convergent. Whether it is or not requires further testing. (c) a = lim a = a = ( )+ lim a =0so again this series may be convergent. Divergence of the harmonic series The harmonic series: has a general term a n = which clearly gets smaller and smaller as n. However, surrisingly, the series is divergent. Its divergence is demonstrated by showing that the harmonic n series is greater than a series which is obviously divergent. We do this by grouing the terms of the harmonic series in a articular way: ( ) ( 3 + ) ( ) + 8 HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series 4
5 Now ( ) ( ) > = ( ) 6 > = > = etc and so on. Hence the harmonic series satisfies: ( ) ( ) ( ) + 8 ( ) > + + ( ) + ( ) + The right-hand side of this inequality is clearly divergent so the harmonic series is divergent Convergence of the alternating harmonic series As with the harmonic series we shall grou the terms of the alternating harmonic series, this time to dislay its convergence. The alternating harmonic series is: S = This series may be re-groued in two distinct ways. st re-grouing ( 5 + = ) ( 3 4 ) ( 5 6 ) 7 each term in brackets is ositive since >, > and so on. So we easily conclude that S< since we are subtracting only ositive numbers from. nd re-grouing ( 5 + = ) ( + 3 ) ( ) + 6 Again, each term in brackets is ositive since >, >, > and so on So we can also argue that S> since we are adding only ositive numbers to the value of the first term,. The conclusion that is forced uon us is that <S< so the alternating series is convergent since its sum, S, lies in the range in Section 6.5 that S =ln It will be shown 5 HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
6 . General Tests for Convergence The techniques we have alied to analyse the harmonic and the alternating harmonic series are one-off :- they cannot be alied to infinite series in general. However, there are many tests that can be used to determine the convergence roerties of infinite series. Of the large number available we shall only consider two such tests in detail. The alternating series test An alternating series is a secial tye of series in which the sign changes from one term to the next. They have the form a a + a 3 a 4 + (in which each a i,i=,, 3,... is a ositive number) Examles are: (a) + + (b) (c) For series of this tye there is a simle criterion for convergence: The alternating series Key Point The Alternating Series Test a a + a 3 a 4 + (in which each a i,i=,, 3,... are ositive numbers) is convergent if and only if the terms continually decrease: the terms decrease to zero: a >a >a 3 >... a 0 as increases (mathematically lim a =0) This is called the alternating series test. Which of the following series are convergent (a) ( ) ( ) ( +) (b) ( ) + HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series 6
7 (a) First, write out the series: Now examine the series for convergence ( ) ( +) = ( ( + as increases. Since the individual terms of the series do not converge ) to zero this is therefore a divergent series. (b) Aly the rocedure used in (a) to roblem (b). This series is an alternating series of the form a a + a 3 a 4 + in which a =. The a sequence is a decreasing sequence since > > 3 >... Also lim ) =0. Hence the series is convergent by the alternating series test. 7 HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
8 3. The Ratio Test This test, which is one of the most useful and widely used convergence tests, alies only to series of ositive terms. Key Point The Ratio Test Let a beaseries of ositive terms. Suose, as increases, the limit of a + a a + anumber λ. That is lim = λ. Then, it is ossible to show that: a equals if λ>, then if λ<, then if λ =, then a diverges a converges a may be convergent or divergent. That is, the test is inconclusive in this case. Examle Use the ratio test to examine the convergence of the series (a) ! 3! 4! (b) +x + x + x 3 + Solution (i) The general term in this series is! i.e. +! + 3! + =! a =! a + = ( + )! and the ratio a + a =! ( + )! = ( )...(3)()() ( +)( )...(3)()() = ( +) HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series 8
9 Solution (contd.) a + lim a = lim ( +) =0 Since 0 < the series is convergent. In fact, it will be easily shown using the techniques outlined in in Section 6.5 that +! + + =e.78 3! (ii) Here we must assume that x>0since we can only aly the ratio test to a series of ositive terms. Now +x + x + x 3 + = x so that and a + lim a a = x, a + = x x = lim = lim x x = x Thus, using the ratio test we deduce that (if x is a ositive number) this series will only converge if x<. We will see in Section 6.4 that +x + x + x 3 + = x rovided 0 <x<. (relace x by x and choose = inthe Binomial series). Use the ratio test to examine the convergence of the series: ln (ln 3) + 7 (ln 3) 3 + First, find the general term of the series. a = = ln 3 (ln 3) 3 so a (ln 3) = 3 (ln 3) Now find a + a + = 9 HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
10 a + = (+)3 (ln 3) + Finally, obtain lim a + a a + a + = lim = a a a+ a = ( + )3 (. Now + (ln 3) )3 (ln 3) < Hence this is a convergent series. = ( + )3 a + as increases lim = a Note that in all of these examles and guided exercises we have decided uon the convergence or divergence of various series; we have not been able to use the tests to discover what actual number the convergent series converges to. 4. Absolute and Conditional Convergence The ratio test alies to series of ositive terms. Indeed this is true of many related tests for convergence. However, as we have seen, not all series are series of ositive terms. To aly the ratio test such series must first be converted into series of ositive terms. This is easily done. Consider two series a and a. The latter series, obviously directly related to the first, is a series of ositive terms. Using imrecise language, it is harder for the second series to converge than it is for the first, since, in the first, some of the terms may be negative and cancel out art of the contribution from the ositive terms. No such cancellations can take lace in the second series since they are all ositive terms. Thus it is lausible that if a converges so does a. This leads to the following definition. Definition Conditional Convergence A convergent series a for which its related series conditionally convergent Absolute Convergence A convergent series a is said to be absolutely convergent if a is divergent is said to be a is convergent. HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series 0
11 For examle, the alternating harmonic series: ( ) + = is conditionally convergent since the series of ositive terms ( ) + = is divergent. Show that the series! + 4! + is absolutely convergent. 6! First, find the general term of the series + + = ( ) a! 4! 6!! + 4! 6! + = ( ) ()! a ( ) ()! The related series of ositive terms is = ( ) a! 4! 6! = Now use the ratio test to examine the convergence of this series th term = ( +) th term = so a = ()! ()! [ ] ( +) th term What is lim? th term [ ] ( +) th term lim = th term th term = ( +) th term = ()! ((+))! HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
12 ()! (( + ))! = ( )... ( + )( + )( )... = ( + )( +) So the series of ositive terms is convergent by the ratio test. Hence convergent. 0as increases. ( ) ()! is absolutely Exercises. Which of the following alternating series are convergent? ( ) ln(3) ( ) + sin( +) π (a) (b) (c) + ( + 00). Use the ratio test to examine the convergence of the series: e 4 3 (a) (b) (c) ( +) +! (d) (0.3) (e) ( ) For what values of x are the following series absolutely convergent? ( ) x ( ) x (a) (b)! Answers. (a) convergent, (b) convergent, (c) divergent. (a) λ = 0so convergent, (b) λ = 0so convergent, (c) λ = so test is inconclusive. However, since > then the given series is divergent by comarison with the harmonic / series. (d) λ = 0/3 sodivergent, (e) Not a series of ositive terms so the ratio test cannot be alied. x 3. (a) The related series of ositive terms is. For this series, using the ratio test we find λ = x so the original series is absolutely convergent if x <. x (b) The related series of ositive terms is.for this series, using the ratio test we find! λ = 0(irresective of the value of x) sothe original series is absolutely convergent for all values of x. HELM (VERSION : March 8, 004): Workbook Level 6.: Infinite Series
16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6.2 Introduction We extend the concet of a finite series, met in Section 6., to the situation in which the number of terms increase without bound. We define what is meant by an infinite
More information16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this Section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
More information16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationAbout the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project
About the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project undertaken by a consortium of five English universities led by
More informationTesting Series with Mixed Terms
Testing Series with Mixed Terms Philippe B. Laval KSU Today Philippe B. Laval (KSU) Series with Mixed Terms Today 1 / 17 Outline 1 Introduction 2 Absolute v.s. Conditional Convergence 3 Alternating Series
More informationSeries Handout A. 1. Determine which of the following sums are geometric. If the sum is geometric, express the sum in closed form.
Series Handout A. Determine which of the following sums are geometric. If the sum is geometric, exress the sum in closed form. 70 a) k= ( k ) b) 50 k= ( k )2 c) 60 k= ( k )k d) 60 k= (.0)k/3 2. Find the
More informationElementary theory of L p spaces
CHAPTER 3 Elementary theory of L saces 3.1 Convexity. Jensen, Hölder, Minkowski inequality. We begin with two definitions. A set A R d is said to be convex if, for any x 0, x 1 2 A x = x 0 + (x 1 x 0 )
More informationStatics and dynamics: some elementary concepts
1 Statics and dynamics: some elementary concets Dynamics is the study of the movement through time of variables such as heartbeat, temerature, secies oulation, voltage, roduction, emloyment, rices and
More informationLecture 6. 2 Recurrence/transience, harmonic functions and martingales
Lecture 6 Classification of states We have shown that all states of an irreducible countable state Markov chain must of the same tye. This gives rise to the following classification. Definition. [Classification
More informationA CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave
More informationTranspose of the Weighted Mean Matrix on Weighted Sequence Spaces
Transose of the Weighted Mean Matri on Weighted Sequence Saces Rahmatollah Lashkariour Deartment of Mathematics, Faculty of Sciences, Sistan and Baluchestan University, Zahedan, Iran Lashkari@hamoon.usb.ac.ir,
More informationAveraging sums of powers of integers and Faulhaber polynomials
Annales Mathematicae et Informaticae 42 (20. 0 htt://ami.ektf.hu Averaging sums of owers of integers and Faulhaber olynomials José Luis Cereceda a a Distrito Telefónica Madrid Sain jl.cereceda@movistar.es
More informationMATH 250: THE DISTRIBUTION OF PRIMES. ζ(s) = n s,
MATH 50: THE DISTRIBUTION OF PRIMES ROBERT J. LEMKE OLIVER For s R, define the function ζs) by. Euler s work on rimes ζs) = which converges if s > and diverges if s. In fact, though we will not exloit
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationApplications to stochastic PDE
15 Alications to stochastic PE In this final lecture we resent some alications of the theory develoed in this course to stochastic artial differential equations. We concentrate on two secific examles:
More informationIntroduction to Banach Spaces
CHAPTER 8 Introduction to Banach Saces 1. Uniform and Absolute Convergence As a rearation we begin by reviewing some familiar roerties of Cauchy sequences and uniform limits in the setting of metric saces.
More informationAn Estimate For Heilbronn s Exponential Sum
An Estimate For Heilbronn s Exonential Sum D.R. Heath-Brown Magdalen College, Oxford For Heini Halberstam, on his retirement Let be a rime, and set e(x) = ex(2πix). Heilbronn s exonential sum is defined
More informationTesting Series With Mixed Terms
Testing Series With Mixed Terms Philippe B. Laval Series with Mixed Terms 1. Introduction 2. Absolute v.s. Conditional Convergence 3. Alternating Series 4. The Ratio and Root Tests 5. Conclusion 1 Introduction
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationTowards understanding the Lorenz curve using the Uniform distribution. Chris J. Stephens. Newcastle City Council, Newcastle upon Tyne, UK
Towards understanding the Lorenz curve using the Uniform distribution Chris J. Stehens Newcastle City Council, Newcastle uon Tyne, UK (For the Gini-Lorenz Conference, University of Siena, Italy, May 2005)
More information23.4. Convergence. Introduction. Prerequisites. Learning Outcomes
Convergence 3.4 Introduction In this Section we examine, briefly, the convergence characteristics of a Fourier series. We have seen that a Fourier series can be found for functions which are not necessarily
More informationCOMMUNICATION BETWEEN SHAREHOLDERS 1
COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov
More informationAbsolute Convergence and the Ratio Test
Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only
More informationMATH 3240Q Introduction to Number Theory Homework 7
As long as algebra and geometry have been searated, their rogress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched
More informationMA3H1 TOPICS IN NUMBER THEORY PART III
MA3H1 TOPICS IN NUMBER THEORY PART III SAMIR SIKSEK 1. Congruences Modulo m In quadratic recirocity we studied congruences of the form x 2 a (mod ). We now turn our attention to situations where is relaced
More information7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =
More informationCybernetic Interpretation of the Riemann Zeta Function
Cybernetic Interretation of the Riemann Zeta Function Petr Klán, Det. of System Analysis, University of Economics in Prague, Czech Reublic, etr.klan@vse.cz arxiv:602.05507v [cs.sy] 2 Feb 206 Abstract:
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More informationPower Series. Part 1. J. Gonzalez-Zugasti, University of Massachusetts - Lowell
Power Series Part 1 1 Power Series Suppose x is a variable and c k & a are constants. A power series about x = 0 is c k x k A power series about x = a is c k x a k a = center of the power series c k =
More informationOn Wald-Type Optimal Stopping for Brownian Motion
J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of
More information10.1 Sequences. A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence.
10.1 Sequences A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence. Notation: A sequence {a 1, a 2, a 3,...} can be denoted
More informationThe Arm Prime Factors Decomposition
The Arm Prime Factors Decomosition Arm Boris Nima arm.boris@gmail.com Abstract We introduce the Arm rime factors decomosition which is the equivalent of the Taylor formula for decomosition of integers
More informationF(p) y + 3y + 2y = δ(t a) y(0) = 0 and y (0) = 0.
Page 5- Chater 5: Lalace Transforms The Lalace Transform is a useful tool that is used to solve many mathematical and alied roblems. In articular, the Lalace transform is a technique that can be used to
More informationReal Analysis 1 Fall Homework 3. a n.
eal Analysis Fall 06 Homework 3. Let and consider the measure sace N, P, µ, where µ is counting measure. That is, if N, then µ equals the number of elements in if is finite; µ = otherwise. One usually
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationHAUSDORFF MEASURE OF p-cantor SETS
Real Analysis Exchange Vol. 302), 2004/2005,. 20 C. Cabrelli, U. Molter, Deartamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and CONICET, Pabellón I - Ciudad Universitaria,
More informationANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM
ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM JOHN BINDER Abstract. In this aer, we rove Dirichlet s theorem that, given any air h, k with h, k) =, there are infinitely many rime numbers congruent to
More informationIntroduction to Probability and Statistics
Introduction to Probability and Statistics Chater 8 Ammar M. Sarhan, asarhan@mathstat.dal.ca Deartment of Mathematics and Statistics, Dalhousie University Fall Semester 28 Chater 8 Tests of Hyotheses Based
More informationp-adic Measures and Bernoulli Numbers
-Adic Measures and Bernoulli Numbers Adam Bowers Introduction The constants B k in the Taylor series exansion t e t = t k B k k! k=0 are known as the Bernoulli numbers. The first few are,, 6, 0, 30, 0,
More informationEigenvalues and Eigenvectors
Contents Eigenvalues and Eigenvectors. Basic Concepts. Applications of Eigenvalues and Eigenvectors 8.3 Repeated Eigenvalues and Symmetric Matrices 3.4 Numerical Determination of Eigenvalues and Eigenvectors
More information8.7 Associated and Non-associated Flow Rules
8.7 Associated and Non-associated Flow Rules Recall the Levy-Mises flow rule, Eqn. 8.4., d ds (8.7.) The lastic multilier can be determined from the hardening rule. Given the hardening rule one can more
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationConvergence of random variables, and the Borel-Cantelli lemmas
Stat 205A Setember, 12, 2002 Convergence of ranom variables, an the Borel-Cantelli lemmas Lecturer: James W. Pitman Scribes: Jin Kim (jin@eecs) 1 Convergence of ranom variables Recall that, given a sequence
More information{ sin(t), t [0, sketch the graph of this f(t) = L 1 {F(p)}.
EM Solved roblems Lalace & Fourier transform c Habala 3 EM Solved roblems Lalace & Fourier transform Find the Lalace transform of the following functions: ft t sint; ft e 3t cos3t; 3 ft e 3s ds; { sint,
More informationSECTION 5: FIBRATIONS AND HOMOTOPY FIBERS
SECTION 5: FIBRATIONS AND HOMOTOPY FIBERS In this section we will introduce two imortant classes of mas of saces, namely the Hurewicz fibrations and the more general Serre fibrations, which are both obtained
More informationSTA 250: Statistics. Notes 7. Bayesian Approach to Statistics. Book chapters: 7.2
STA 25: Statistics Notes 7. Bayesian Aroach to Statistics Book chaters: 7.2 1 From calibrating a rocedure to quantifying uncertainty We saw that the central idea of classical testing is to rovide a rigorous
More informationElliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015
18.783 Ellitic Curves Sring 2015 Problem Set #1 Due: 02/13/2015 Descrition These roblems are related to the material covered in Lectures 1-2. Some of them require the use of Sage, and you will need to
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers:,2,..., These are constructed using Peano axioms. We will not get into the hilosohical questions related to this and simly assume the
More informationLecture: Condorcet s Theorem
Social Networs and Social Choice Lecture Date: August 3, 00 Lecture: Condorcet s Theorem Lecturer: Elchanan Mossel Scribes: J. Neeman, N. Truong, and S. Troxler Condorcet s theorem, the most basic jury
More informationCSC165H, Mathematical expression and reasoning for computer science week 12
CSC165H, Mathematical exression and reasoning for comuter science week 1 nd December 005 Gary Baumgartner and Danny Hea hea@cs.toronto.edu SF4306A 416-978-5899 htt//www.cs.toronto.edu/~hea/165/s005/index.shtml
More informationMath 104B: Number Theory II (Winter 2012)
Math 104B: Number Theory II (Winter 01) Alina Bucur Contents 1 Review 11 Prime numbers 1 Euclidean algorithm 13 Multilicative functions 14 Linear diohantine equations 3 15 Congruences 3 Primes as sums
More informationSets of Real Numbers
Chater 4 Sets of Real Numbers 4. The Integers Z and their Proerties In our revious discussions about sets and functions the set of integers Z served as a key examle. Its ubiquitousness comes from the fact
More informationHENSEL S LEMMA KEITH CONRAD
HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar
More informationThe Longest Run of Heads
The Longest Run of Heads Review by Amarioarei Alexandru This aer is a review of older and recent results concerning the distribution of the longest head run in a coin tossing sequence, roblem that arise
More informationt 0 Xt sup X t p c p inf t 0
SHARP MAXIMAL L -ESTIMATES FOR MARTINGALES RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI ABSTRACT. Let X be a suermartingale starting from 0 which has only nonnegative jums. For each 0 < < we determine the best
More informationPractice Final Solutions
Practice Final Solutions 1. True or false: (a) If a is a sum of three squares, and b is a sum of three squares, then so is ab. False: Consider a 14, b 2. (b) No number of the form 4 m (8n + 7) can be written
More informationInfinitely Many Quadratic Diophantine Equations Solvable Everywhere Locally, But Not Solvable Globally
Infinitely Many Quadratic Diohantine Equations Solvable Everywhere Locally, But Not Solvable Globally R.A. Mollin Abstract We resent an infinite class of integers 2c, which turn out to be Richaud-Degert
More informationIMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES
IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES OHAD GILADI AND ASSAF NAOR Abstract. It is shown that if (, ) is a Banach sace with Rademacher tye 1 then for every n N there exists
More informationMATH342 Practice Exam
MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June 2012. It is not imlied that all questions on the real examination will follow the content of the ractice
More informationSession 5: Review of Classical Astrodynamics
Session 5: Review of Classical Astrodynamics In revious lectures we described in detail the rocess to find the otimal secific imulse for a articular situation. Among the mission requirements that serve
More informationDiophantine Equations and Congruences
International Journal of Algebra, Vol. 1, 2007, no. 6, 293-302 Diohantine Equations and Congruences R. A. Mollin Deartment of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada,
More information0.6 Factoring 73. As always, the reader is encouraged to multiply out (3
0.6 Factoring 7 5. The G.C.F. of the terms in 81 16t is just 1 so there is nothing of substance to factor out from both terms. With just a difference of two terms, we are limited to fitting this olynomial
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationMTH 3102 Complex Variables Practice Exam 1 Feb. 10, 2017
Name (Last name, First name): MTH 310 Comlex Variables Practice Exam 1 Feb. 10, 017 Exam Instructions: You have 1 hour & 10 minutes to comlete the exam. There are a total of 7 roblems. You must show your
More informationSums of independent random variables
3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where
More informationFeedback-error control
Chater 4 Feedback-error control 4.1 Introduction This chater exlains the feedback-error (FBE) control scheme originally described by Kawato [, 87, 8]. FBE is a widely used neural network based controller
More informationRECIPROCITY LAWS JEREMY BOOHER
RECIPROCITY LAWS JEREMY BOOHER 1 Introduction The law of uadratic recirocity gives a beautiful descrition of which rimes are suares modulo Secial cases of this law going back to Fermat, and Euler and Legendre
More informationarxiv:cond-mat/ v2 25 Sep 2002
Energy fluctuations at the multicritical oint in two-dimensional sin glasses arxiv:cond-mat/0207694 v2 25 Se 2002 1. Introduction Hidetoshi Nishimori, Cyril Falvo and Yukiyasu Ozeki Deartment of Physics,
More informationIntroduction to Group Theory Note 1
Introduction to Grou Theory Note July 7, 009 Contents INTRODUCTION. Examles OF Symmetry Grous in Physics................................. ELEMENT OF GROUP THEORY. De nition of Grou................................................
More information1 Riesz Potential and Enbeddings Theorems
Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for
More informationx 2 a mod m. has a solution. Theorem 13.2 (Euler s Criterion). Let p be an odd prime. The congruence x 2 1 mod p,
13. Quadratic Residues We now turn to the question of when a quadratic equation has a solution modulo m. The general quadratic equation looks like ax + bx + c 0 mod m. Assuming that m is odd or that b
More informationk- price auctions and Combination-auctions
k- rice auctions and Combination-auctions Martin Mihelich Yan Shu Walnut Algorithms March 6, 219 arxiv:181.3494v3 [q-fin.mf] 5 Mar 219 Abstract We rovide for the first time an exact analytical solution
More informationarxiv: v1 [math.nt] 11 Jun 2016
ALMOST-PRIME POLYNOMIALS WITH PRIME ARGUMENTS P-H KAO arxiv:003505v [mathnt Jun 20 Abstract We imrove Irving s method of the double-sieve [8 by using the DHR sieve By extending the uer and lower bound
More informationarxiv: v4 [math.nt] 11 Oct 2017
POPULAR DIFFERENCES AND GENERALIZED SIDON SETS WENQIANG XU arxiv:1706.05969v4 [math.nt] 11 Oct 2017 Abstract. For a subset A [N], we define the reresentation function r A A(d := #{(a,a A A : d = a a }
More informationPretest (Optional) Use as an additional pacing tool to guide instruction. August 21
Trimester 1 Pretest (Otional) Use as an additional acing tool to guide instruction. August 21 Beyond the Basic Facts In Trimester 1, Grade 7 focus on multilication. Daily Unit 1: The Number System Part
More informationNumerical Linear Algebra
Numerical Linear Algebra Numerous alications in statistics, articularly in the fitting of linear models. Notation and conventions: Elements of a matrix A are denoted by a ij, where i indexes the rows and
More information14.4. Lengths of Curves and Surfaces of Revolution. Introduction. Prerequisites. Learning Outcomes
Lengths of Curves and Surfaces of Revolution 4.4 Introduction Integration can be used to find the length of a curve and the area of the surface generated when a curve is rotated around an axis. In this
More informationChapter 2 Arithmetic Functions and Dirichlet Series.
Chater 2 Arithmetic Functions and Dirichlet Series. [4 lectures] Definition 2.1 An arithmetic function is any function f : N C. Examles 1) The divisor function d (n) (often denoted τ (n)) is the number
More informationON FREIMAN S 2.4-THEOREM
ON FREIMAN S 2.4-THEOREM ØYSTEIN J. RØDSETH Abstract. Gregory Freiman s celebrated 2.4-Theorem says that if A is a set of residue classes modulo a rime satisfying 2A 2.4 A 3 and A < /35, then A is contained
More information#A6 INTEGERS 15A (2015) ON REDUCIBLE AND PRIMITIVE SUBSETS OF F P, I. Katalin Gyarmati 1.
#A6 INTEGERS 15A (015) ON REDUCIBLE AND PRIMITIVE SUBSETS OF F P, I Katalin Gyarmati 1 Deartment of Algebra and Number Theory, Eötvös Loránd University and MTA-ELTE Geometric and Algebraic Combinatorics
More informationThe Second Law: The Machinery
The Second Law: The Machinery Chater 5 of Atkins: The Second Law: The Concets Sections 3.7-3.9 8th Ed, 3.3 9th Ed; 3.4 10 Ed.; 3E 11th Ed. Combining First and Second Laws Proerties of the Internal Energy
More information2.4. Characterising functions. Introduction. Prerequisites. Learning Outcomes
Characterising functions 2.4 Introduction There are a number of different terms used to describe the ways in which functions behave. In this section we explain some of these terms and illustrate their
More informationA generalization of Amdahl's law and relative conditions of parallelism
A generalization of Amdahl's law and relative conditions of arallelism Author: Gianluca Argentini, New Technologies and Models, Riello Grou, Legnago (VR), Italy. E-mail: gianluca.argentini@riellogrou.com
More informationdn i where we have used the Gibbs equation for the Gibbs energy and the definition of chemical potential
Chem 467 Sulement to Lectures 33 Phase Equilibrium Chemical Potential Revisited We introduced the chemical otential as the conjugate variable to amount. Briefly reviewing, the total Gibbs energy of a system
More informationFormula-Preferential Systems for Paraconsistent Non-Monotonic Reasoning (an extended abstract)
Formula-Preferential Systems for Paraconsistent Non-Monotonic Reasoning (an extended abstract) Abstract We rovide a general framework for constructing natural consequence relations for araconsistent and
More informationMathematics as the Language of Physics.
Mathematics as the Language of Physics. J. Dunning-Davies Deartment of Physics University of Hull Hull HU6 7RX England. Email: j.dunning-davies@hull.ac.uk Abstract. Courses in mathematical methods for
More informationMATH 361: NUMBER THEORY ELEVENTH LECTURE
MATH 361: NUMBER THEORY ELEVENTH LECTURE The subjects of this lecture are characters, Gauss sums, Jacobi sums, and counting formulas for olynomial equations over finite fields. 1. Definitions, Basic Proerties
More informationConvex Optimization methods for Computing Channel Capacity
Convex Otimization methods for Comuting Channel Caacity Abhishek Sinha Laboratory for Information and Decision Systems (LIDS), MIT sinhaa@mit.edu May 15, 2014 We consider a classical comutational roblem
More informationOn Doob s Maximal Inequality for Brownian Motion
Stochastic Process. Al. Vol. 69, No., 997, (-5) Research Reort No. 337, 995, Det. Theoret. Statist. Aarhus On Doob s Maximal Inequality for Brownian Motion S. E. GRAVERSEN and G. PESKIR If B = (B t ) t
More information2.4. Characterising Functions. Introduction. Prerequisites. Learning Outcomes
Characterising Functions 2.4 Introduction There are a number of different terms used to describe the ways in which functions behave. In this Section we explain some of these terms and illustrate their
More informationBEST POSSIBLE DENSITIES OF DICKSON m-tuples, AS A CONSEQUENCE OF ZHANG-MAYNARD-TAO
BEST POSSIBLE DENSITIES OF DICKSON m-tuples, AS A CONSEQUENCE OF ZHANG-MAYNARD-TAO ANDREW GRANVILLE, DANIEL M. KANE, DIMITRIS KOUKOULOPOULOS, AND ROBERT J. LEMKE OLIVER Abstract. We determine for what
More informationSolved Problems. (a) (b) (c) Figure P4.1 Simple Classification Problems First we draw a line between each set of dark and light data points.
Solved Problems Solved Problems P Solve the three simle classification roblems shown in Figure P by drawing a decision boundary Find weight and bias values that result in single-neuron ercetrons with the
More informationAbsolute Convergence and the Ratio & Root Tests
Absolute Convergence and the Ratio & Root Tests Math114 Department of Mathematics, University of Kentucky February 20, 2017 Math114 Lecture 15 1/ 12 ( 1) n 1 = 1 1 + 1 1 + 1 1 + Math114 Lecture 15 2/ 12
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More information6.2. The Hyperbolic Functions. Introduction. Prerequisites. Learning Outcomes
The Hyperbolic Functions 6. Introduction The hyperbolic functions cosh x, sinh x, tanh x etc are certain combinations of the exponential functions e x and e x. The notation implies a close relationship
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More informationInfinite Number of Twin Primes
dvances in Pure Mathematics, 06, 6, 95-97 htt://wwwscirorg/journal/am ISSN Online: 60-08 ISSN Print: 60-068 Infinite Number of Twin Primes S N Baibeov, Durmagambetov LN Gumilyov Eurasian National University,
More information