# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

4 Pages

### midterm2-solution

Course: MATH 140a, Fall 2008
School: UC Irvine
Rating:

Word Count: 851

#### Document Preview

midterm Second exam - 140A Solutions March, 5. 2007 1 Exercise 1 (25 points) s2 +1 n 2. Let us consider the sequence (sn ) for n N dened as s1 = 2 and sn+1 = (10 points) Assuming lim sn = s exists. Show that s = 1. (10 points) Show that sn is non-decreasing. (5 points) Does the limit of the sequence actually exist ? (explain why) Solution: Let lim sn = s exists. Therefore lim s2 = s2 and n lim sn+1 = lim...

Register Now

#### Unformatted Document Excerpt

Coursehero >> California >> UC Irvine >> MATH 140a

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
midterm Second exam - 140A Solutions March, 5. 2007 1 Exercise 1 (25 points) s2 +1 n 2. Let us consider the sequence (sn ) for n N dened as s1 = 2 and sn+1 = (10 points) Assuming lim sn = s exists. Show that s = 1. (10 points) Show that sn is non-decreasing. (5 points) Does the limit of the sequence actually exist ? (explain why) Solution: Let lim sn = s exists. Therefore lim s2 = s2 and n lim sn+1 = lim Solving for s leads to s = 1. Let us compute sn+1 sn sn+1 sn = s2...
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

UC Irvine - MATH - 140a
Chapter 1 Axioms of the Real Number System1.1 Introductory Remarks: What constitutes a proof ?One of the hurdles for a student encountering a rigorous calculus course for the rst time, is what level of detail is expected in a proof. If every state
UC Irvine - MATH - 141
; ; ; ; file init.scm ; ; ICS 22 Scheme lab assignment file ; ; ; ; Revised and tested for Windows NT by ; ; Li-Wei (Gary) Chen ; ; Summer 1998 ; ;; ; Redefine &quot;car&quot; and &quot;cdr&quot; for clarity (define first car) (define rest cdr) ; Define atom? in terms o
UC Irvine - MATH - 146
UC Irvine - MATH - 150
BME150 BIOLOGICAL MASS TRANSFER (Required for BME and BMEP) Catalog Data: BME150 Biological Mass Transfer (Credit Units: 4) Mass transfer in gas, liquid and solid with application to biological systems. Free and facilitated diffusion, active transpor
UC Irvine - MATH - 150
BME150 BIOLOGICAL MASS TRANSFER (Required for BME and BMEP) Catalog Data: BME150 Biological Mass Transfer (Credit Units: 4) Mass transfer in gas, liquid and solid with application to biological systems. Free and facilitated diffusion, active transpor
UC Irvine - MATH - 152
Physica D 152153 (2001) 505519The NavierStokes-alpha model of uid turbulenceCiprian Foias a, , Darryl D. Holm b , Edriss S. Titi ccDepartment of Mathematics, Indiana University, Bloomington, IN 47405, USA T-Division and CNLS, MS-B284, Los Alamo
UC Irvine - MATH - 161
CEE 161 WATER AND WASTEWATER TREATMENT (Required for CE; Elective for EnE) Catalog Data: CEE 161: Water And Wastewater Treatment (Credit Units: 4) S. Water quality parameters. Water use, reclamation, and reuse. Introduction to modeling and designing
UC Irvine - MATH - 161
ICS 161Spring 2007 Design and Analysis of Algorithms George Lueker Summing Factors Recurrences will come up a number of times in this course, but we dont have the time to provide a thorough introduction to the subject in class. (In Math 6A you have p
UC Irvine - MATH - 171a
MATH 171 A, 44365 Linear Programming, Summer 2005 Instructor: Kai Huang E-mail: khuang@math.uci.edu Oce/Hours: MSTB 278; M W 4:30-5:30 PM. Meeting times and Places: M Tu W Th 3:00-4:20 pm, IERF 101 (Lec.) Textbook: Bernard Kolman and Robert E.
UC Irvine - MATH - 180
MATH 180, FINALMarch 19, 2007 Answers1. (a) Text, p. 79 (b) Text, p. 69 2. Text, Theorem 6.2 3. Suppose p = 2 and p = 7. Then 7 is a square modulo p if and only if 7 = 1, and p7 p=1 p7 p= (1)p1 2(1)p1 71 2 2( p = (1)2(p1) ( p = (
UC Irvine - MATH - 180
Math 180, problem set #7due March 6, 2007 (1) Compute the continued fraction for 105/38. Using this, nd integers x and y such that 105x + 38y = 1. (2) Compute the continued fraction for 95/43. Using this, nd integers x and y such that 95x + 43y = 1.
UC Irvine - MATH - 180
Math 180, problem set #8due March 13, 2007 (1) There is a rational number r, with denominator less than 50, whose decimal expansion begins 1.2941176470588 . . . Express r as a fraction. (You may need to use a calculator to compute a continued fracti
UC Irvine - MATH - 190w
ENGR 190W COMMUNICATIONS IN THE PROFESSIONAL WORLD (Required for ChE, CpE, EE, and MSE) Catalog Data: ENGR 190W: Communications in the Professional World (Credit Units: 4) F, W, S, Summer. Workshop in technical and scientific writing. Oral presentati
UC Irvine - MATH - 194
Math 194, problem set #1For discussion Tuesday October 91. Show that some multiple of 1232123432123454321 contains all 10 digits (at least once) when written in base 10. Solution 1: Let n = 1232123432123454321. Consider the n consecutive integers
UC Irvine - MATH - 1a
BRIDGESRafael Moran San Diego State University MSPDesign It is the essence of engineering! the process of devising a system, component or process to meet desired needs. It is a decision-making process, in which the basic sciences, mathematics, a
UC Irvine - MATH - 1b
1Chapter 1Chapter 1 Networks, Ethnography, and EmergenceThe question posed in this introductory chapter is a general one: How do new ways of thinking about networks increase our understanding of theoretical and ethnographic problems in the socia
UC Irvine - MATH - 205b
SOLUTIONS FOR MATH 205B MIDTERM1 (15 points): Dene a function f on [0, 1] by setting f (x) = x x is rational . 0 x is irrationalShow that f is not of bounded variation. It suces to nd, for any N N, points x0 &lt; x1 &lt; . . . &lt; xn in [0, 1] s.t. n k=
UC Irvine - MATH - 205b
MATH 205B PRACTICE PROBLEMS FOR THE FINAL1. Determine whether the following series converge or diverge:(a)n=1n n+1n(n+1)(b)n=1lnln(n + 2) . ln(n + 1)2. (a) Suppose f is a Riemann integrable function on [0, 1]. Compute1 nlimx
UC Irvine - MATH - 205b
MATH 205B PRACTICE PROBLEMS FOR THE MIDTERM1. Suppose f has a derivative (nite or innite) everywhere on R, and f (0) = 0. Prove that for any &gt; 0 there exists x R s.t. 0 &lt; |x| &lt; , and |f (x)| &lt; . 2. Suppose f (x) = 1 x=0 , (x) = 0 x=0 0 x0 . 1 x&gt;0
UC Irvine - MATH - 205c
SOLUTIONS FOR MATH 205C FINAL (JUNE 2008)(b) Suppose, for n N, Sn is a closed bounded set of Jordan content zero, and S = nN Sn is bounded. Is it true that S has Jordan content zero? (a) Sn has Jordan content zero, hence measure zero. Therefore, S
UC Irvine - MATH - 205c
MATH 205C PRACTICE PROBLEMS FOR THE FINAL Last update: 06/02/08 1. Problem 14.8. 2. Problem 14.9. 3. Problem 14.11 (assume S is compact). 4. Suppose A and B are bounded Jordan measurable subsets of Rn . Prove that Ac (the complement of A, A B, and A
UC Irvine - MATH - 205c
SOLUTIONS FOR MATH 205C MIDTERM1 (20 points): For each of the following series, determine all the real values of x for which the series converges. (a) k=0xk /2k . Here, ab equals a(b ) , not (ab )c = abc .c c2First compute the radius of con
UC Irvine - MATH - 227a
MATH 227A Fall, 2007Dpp Gradients in Drosophila Wing Discs*(Review Version)December 28, 2007Frederic Y.M. Wan Mathematics University of California, Irvine* Supported by NIH Grants R01-GM67247 R01-GM075309 and P50-GM066051Principal Referenc
UC Irvine - MATH - 227b
MATH 227BMATHEMATICAL BIOLOGY IIWinter, 2008 (January 13, 2008)Lecture A MWF 12:00 - 12:50 p.m.; PSCB 230 Code: 45435 Instructor: Professor Frederic Y.M. Wan (MSTB 267, 824-5529, fwan@uci.edu) Office Hours: Th. 2:303:30 p.m. &amp; Mon 1011 a.m. (by
UC Irvine - MATH - 230b
Math 230B nalDue: March 23, 2005, 4:00pm Turn in your completed exam to Jennifer Dugan in MSTB 103. You may use your class notes and the portions of the text that were covered in class. No other sources are permitted. No collaboration is permitted.
UC Irvine - MATH - 230b
Math 230B, problem set #3 (1) Determine Gal(Q( 6, 11)/Q). What are the automorphisms, and which familiar group is it? (2) Determine the Galois group of x3 7x + 3 Q[x]. Which familiar group is it? (3) If p is prime, show that the Galois group of xp
UC Irvine - MATH - 230c
Math 230C, problem set #2due April 20 (1) Suppose : G GLn (k) is a representation. Show that the map g det(g) is a 1-dimensional representation of G. (2) Suppose : G GLn (C) is a representation. Show that for every g G, (g) is diagonalizable.
UC Irvine - MATH - 230c
Math 230C, problem set #3due April 29 (1) Write down the character table of A4 (i.e., list all irreducible characters and their values on all conjugacy classes). (2) Write down the character table of D10 . (3) If R is a commutative ring, I R is an
UC Irvine - MATH - 230c
Name:Math 230C nal, with solutionsJune 15, 2005, 1:30-3:30pm Closed book, no notes or other aids. Justify your answers carefully and completely. Use the back of the page if necessary, and there is a blank page at the end for extra space. There are
UC Irvine - MATH - 230c
Math 230C, problem set #4due May 13 (1) Rotman #9.70 (2) Rotman #9.86 (3) Rotman #9.89 (4) Rotman #9.90 (5) Rotman #9.95 (6) Suppose that R is a commutative ring and M is a cyclic R-module. (a) Show that the tensor algebra T (M ) is isomorphic to th
UC Irvine - MATH - 232b
Math 232BProblem set #4, due February 20, 2008 (1) Suppose F/K is a nite extension of local elds. Prove that NF/K F and NF/K UF are both open and closed in K . (2) Using the correspondence between quadratic extensions of Qp and subgroups of index
UC Irvine - MATH - 232b
Math 232BProblem set #5, due March 3, 2008 (1) Suppose K is a local eld, O is its ring of integers, p is the maximal ideal of O, k = O/p is the residue eld, p is the characteristic of k, and F is a formal group over O. For every n 1, let F (pn ) de
UC Irvine - MATH - 232b
Math 232BProblem set #1, due January 23, 2008 (1) Suppose K F , K L are number elds, and p is a prime of K. (a) Show that if p splits completely in F and L, then p splits completely in F L. (b) Show that p splits completely in F if and only if p s
UC Irvine - MATH - 232b
Math 232BProblem set #6, due March 12, 2008 (1) Suppose K is a local eld, and let be the completion of an algrbraic closure K. Suppose F1 , F2 are (possibly innite) algebraic extensions of K in K. Show that if the completions of F1 and F2 are equ
UC Irvine - MATH - 232c
Math 232CProblem set #2, due April 16, 2008 (1) Suppose K is a eld of characteristic p, K sep is its separable closure, and G = Gal(K sep /K). (a) Show that the map : K sep K sep dened by (x) = xp x is a surjective homomorphism of G-modules, with
UC Irvine - MATH - 232c
NOTES ON MATH 232C: ALGEBRAIC NUMBER THEORY (8/)Abstract. These notes are based on my notes scribed for the course, Algebraic Number Theory, oered at UCI, lectured by Prof. Daqing Wan. Any mistakes in the notes are solely due to me. Please email me
UC Irvine - MATH - 232c
NOTES ON MATH 232C: ALGEBRAIC NUMBER THEORY (24/)Abstract. These notes are based on my notes scribed for the course, Algebraic Number Theory, oered at UCI, lectured by Prof. Daqing Wan. Any mistakes in the notes are solely due to me.Sk (g) =q k1
UC Irvine - MATH - 232c
NOTES ON MATH 232C: ALGEBRAIC NUMBER THEORY (5/)Abstract. These notes are based on my notes scribed for the course, Algebraic Number Theory, oered at UCI, lectured by Prof. Daqing Wan. Any mistakes in the notes are solely due to me. Please email me
UC Irvine - MATH - 234a
Topics in Algebra: Zeta Functions of Toric Hypersurfaces (p-Adic Theory) Daqing WanContentsList of Figures Zeta Functions and -regularity Introduction 1. Polytope constructions 1.1. The Ehrhart Polynomial 1.2. The Algebra S 1.3. Proof of Ehrhart 2
UC Irvine - MATH - 270a
Notes 8: Predicate logic and inferenceICS 270a Spring 2003Outline New ontologyobjects,relations,properties,functions. Constants, predicates,properties,functions meaning of new syntax Resolution Forward-chaining, Backword-chaining unification
UC Irvine - MATH - 3a
Name ID Quiz 4 Math 3a May 11, 2006 Of the two following maps, one is a linear transformation and the other is not. Determine which is and which is not and justify your conclusion. For the one that is a linear transformation nd the matrix representat
UC Irvine - MATH - 3a
Name ID Quiz 5 Math 3a May 18, 2006 Let x = ( 22 , 22 )T , and y = (1, 1)T . 1)Compute the lengths of x and y, the distance between x and y and the scalar product of x and y. |x| = |y| = x y=T 22 ) + ( 22 )2 = 1 2 2 + (1)2 = 1 2 (|x y| =
UC Irvine - MATH - 3a
Name ID Quiz 2 Math 3a April 20, 2006 Consider the vectors (0, 0, 2)t , (1, 1, 2)t , (2, 1, 2)t . Show that these three vectors span R3 . Also show the vector (7, 6, 18)t can be made as a linear combination of these three vectors. (HINT: The coefcent
UC Irvine - MATH - 3a
3A WORKSHEETAbstract. Some of these questions have been phrased somewhat ambiguously. Please take care to think about each of these problems. Each question was designed to illustrate some phenomenon or detail of the subject matter. Feel free to dis
UC Irvine - MATH - 5
Strong Induction&quot;Normal&quot; Induction: If we prove that 1) P(n0) is true for some n0 (typically 0 or 1), and 2) If P(k) is true for any kn0, then P(k+1) is also true. Then P(n) is true for all nn0. &quot;Strong&quot; Induction: If we prove that 1) Q(n0) is true
UC Irvine - MATH - 5
Research Process and ResourcesKathryn Kjaer, kkjaer@uci.edu UCI Physics and Math Librarian May 5, 2005 For Physics 197: Senior Projects and Thesis Prof. Roger McWilliamsScientific communication Scientists document and share their research findings
UC Irvine - MATH - 5
UCI Cal Teach E-News Tuesday, February 5, 2008 Don't forget to get out there and vote today! &quot;You must be the change you wish to see in the world. - Mohandas Gandhi (1869-1948) Appeal of Challenge to 'No Child' Law http:/www.nytimes.com/2008/02/02/us
UC Irvine - MATH - 5
UC Irvine - MATH - 67
Statistics 67 Introduction to Probability and Statistics for Computer ScienceWhen: MWF 12:00noon-12:50pm Where: Physical Sciences Classroom Building 140 (building 413 on campus map) Discussions: W 9:00am-9:50am in SE2 1306 (building 215) W 1:00pm-1:
UC Irvine - MATH - 67
Statistics 67 Introduction to Probability and Statistics for Computer ScienceLecture notes for Statistics Hal Stern University of California, Irvine sternh@uci.edu1From Probability . To this point probability as a measure of uncertainty prob
UC Irvine - MATH - 67
UC Irvine - MATH - 67
PHYSICAL REVIEW E 76, 011601 2007Linear stability analysis for step meandering instabilities with elastic interactions and Ehrlich-Schwoebel barriers1Dong-Hee Yeon,1 Pil-Ryung Cha,2 John S. Lowengrub,3 Axel Voigt,4 and K. Thornton1,*Department
UC Irvine - MATH - 6d
IntegersNumber Theory = Properties of Integers (For this part, assume all values are integers.) &quot;a|b&quot; = &quot;a divides b&quot; = n Z (b=na) &quot;b is a multiple of a.&quot; &quot;a is a factor of b.&quot; &quot;Multiple&quot; always means &quot;integer multiple&quot; Thrm: If a|b and a|c, then a|
UC Irvine - MATH - 6d
Sets&quot;Set&quot;=Unordered collection of Objects &quot;Set Elements&quot; = &quot;Members of the Set&quot; = Objects in Set X={2,3,5,7,11,13} 3X [&quot;3 is a member of X&quot;] 4X [&quot;4 is not a member of X&quot;]Common sets of numbers: N, Z, Q, R Sets are &quot;equal&quot; if and only if (iff) they
UC Irvine - MATH - 6d
Counting: Product and Sum RulesProduct Rule: Assuming a) We need to perform procedure 1 AND procedure 2. b) There are n1 ways to perform procedure 1 and c) n2 ways to perform procedure 2. There are n1n2 ways to perform procedure 1 AND procedure 2. S
UC Irvine - MATH - 71
A New Method for Simulating Strongly Anisotropic Cahn-Hilliard EquationsS. Torabi , S. Wise , J. Lowengrub University of California, Irvine, California, USA A. Rtz, A. Voigt Institute fr Wissenschaftliches Rechnen, Technische Universitt Dresden, Dre
UC Irvine - MGMT - 10
ICS 52: Introduction to Software EngineeringFall Quarter 2004 Professor Richard N. Taylor Lecture Notes: CM, Management, and EvolutionSeveral Illustrations from Ian Sommervilles text http:/www.ics.uci.edu/~taylor/ICS_52_FQ04/syllabus.htmlCopyright
UC Irvine - MGMT - 10
ICS 52: Introduction to Software EngineeringFall Quarter 2002 Professor Richard N. Taylor Lecture Notes: CM, Management, and Evolution Many slides taken from Ian Sommervilles text http:/www.ics.uci.edu/~taylor/ICS_52_FQ02/syllabus.htmlCopyright 202
UC Irvine - MGMT - 10
ICS 52: Introduction to Software EngineeringWinter Quarter 2004 Professor Richard N. Taylor Lecture Notes: CM, Management, and Evolution Many slides taken from Ian Sommervilles texthttp:/www.ics.uci.edu/~taylor/ICS_52_WQ04/syllabus.htmlCopyright 2
UC Irvine - MGMT - 5