Assignment 5 Remarks and Partial Solutions 14.1 Determine which of the following series converge. Justify your answers. (a) (Hueristics: The exponential is the dominant term and will overcome any power. 1 Therefore, we expect this to behave more like 2n (
Homework #6
14.3 Determine (a) (c) (e) which of the following series converge. Justify your answers. 1/ n! (b) (2 + cos n)/3n 1/(2n + n) (d) (1/2)n (50 + 2/n) sin(n/9) (f) (100)n /n! n! (n + 1)!
Solution: (a): Applying the Ratio Test, we obtain lim sup 1/
Continuity and Limits of Functions Exercise Answers
1. Let f be given by f (x) = x R. (a) dom(f + g) = dom(f g) = (, 4], dom(f g) = [2, 2] and dom(g f ) = (, 4] (b) (f g)(0) = 2, (g f )(0) = 4, (f g)(1) = 3,(g f )(1) = 3, (f g)(2) = 0 and (g f )(2) = 2. (
CONTINUITY
Problem 17.4: Prove that the function Proof. Given
x is continuous on its domain [0, ).
> 0, we need to nd a > 0 such that |x xo | < implies |f (x) f (xo )| < . x xo |x xo | < . |f (x) f (xo )| = | x xo | = < xo xo x + xo ()
We want |f (x) f (
Homework 2 Key Answers
140B
1. (3 points each) For each of the following power series, nd the radius of convergence and the interval of convergence: n nx , 1) 2) 3) 1n x, nn 3n 2n+1 x n
.
Solution: (a) For the rst series, let us compute: = lim sup |an |1/
MA250 Test summary
Time: Wednesday, February 25, 2009, (5:30-6:50 p.m.) Place: BA211 (Bricker Academic) All answers require full justication. There will be proofs. The test covers Sections 1-5, and 7-11 from the texbook, plus the Lecture Notes posted on
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a MA238 Winter 2009 - Quiz 2a
Total marks is . Answer all questions in the spaces provided. Calculators are not permitted.
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MA251 Set Theory
Assigned Problems (No. 5)
1. Prove that the collection O of all ordinal numbers does not form a
set.
2. Let P, be the well-ordered set with the usual order. Prove the
Principle of Strong Mathematical Induction:
Let S be a subset of P with
MATH 104, SUMMER 2006, HOMEWORK 5 SOLUTION
BENJAMIN JOHNSON Due July 19
Assignment: Section 14: 14.4, 14.7, 14.10, 14.13(d) Section 15: 15.4, 15.6, 15.7 Section 14 14.4 Determine which of the following series converge. Justify your answer. 1 (a) 2 [n+(1)n
Student ID:
MA238 Winter 2009 Quiz 3
Total marks is . Answer all questions in the spaces provided. Calculators are not permitted.
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[ marks] 1. (a) What is the denition of an Euler path?
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MA251 Set Theory
Assigned Problems (No. 5)
1. Using two different methods to prove each of the following arithmetic
properties of real numbers.
(1) For any R, there is a unique R such that 0. Then is
denoted by .
(2) Using (1) define the subtraction of re
MA250 Lecture Notes: Subsequences and the Heine-Borel Theorem
Manuele Santoprete February 9, 2009
1
Subsequences
An useful criterion to show that a sequence does not converge is given by the following Corollary to Theorem 11.2 Corollary 1. Is (sn ) has tw
MA250 Final Exam summary
Time: Monday, April 27 , 2009, (6:30-8:30 p.m.) Place: AC (Athletic Complex) All answers require full justification. There will be proofs. The test covers Sections 1-5, 7-12, 14, 17-20, 23-25 from the texbook, plus the Lecture No
Math 312, Intro. to Real Analysis: Homework #6 Solutions
Stephen G. Simpson Friday, April 10, 2009
The assignment consists of Exercises 17.3(a,b,c,f), 17.4, 17.9(c,d), 17.10(a,b), 17.14, 18.5, 18.7, 19.1, 19.2(b,c), 19.5 in the Ross textbook. Each exercis
Homework 1
1.4. (a) When we evaluate the sum f (n) = 1 + 3 + + (2n 1) for low values of n we obtain f (1) = 1, f (2) = 4, f (3) = 9 and f (4) = 16. From this data, it seems reasonable to guess that f (n) = n2 . (b) Let Pn be the statement 1 + 3 + + (2n 1)
Math 125a homework 1
Winter 2009
Problems: 1) 17.3 a) b) 2) 17.4: prove using the denition of continuity and hint in the book. 3) 17.9 a) 4) 17.10 a) 5) 17.11: Hint: One direction of the if and only if statement fallows directly form the denition of conti
Solutions to Homework 4
February 9, 2009
20.10 Prove that a:limx f (x) = , b:limx0 f (x) = , c: limx0+ f (x) = , d: 2 limx0 f (x) Does not exists, and e: limx f (x) = for f (x) = 1x . You can do this by x - or through the sequence denition. Proof: a: Let
Solutions to Homework 2
February 5, 2009
19.2(b) Prove that f (x) = x2 is uniformly continuous on [0, 3] using the denition:
Proof: Note that |x + y | 6 for all x, y [0, 3] by the triangle inequality. For epsilon > 0 let = 6 so that for all |x y | < and
Solutions to Homework 2
February 2, 2009
18.4: Let S R and suppose a sequence (xn ) in S that converges to a number x0 S . Show / that there an unbounded continuous function on S . Note: In other words, we want to show that there is a function f dened on
MATH 104, SUMMER 2006, HOMEWORK 9 SOLUTION
BENJAMIN JOHNSON Due August 7
Assignment: Section 25: 25.3, 25.15 Section 26: 26.2, 26.5 Section 28: 28.2(a), 28.6(b), 28.8 Section 25 25.3 Let fn ( x) = 2n+cos2xx for all real numbers x. n+sin (a) Show that fn c
MATH 104, SUMMER 2006, HOMEWORK 8 SOLUTION
BENJAMIN JOHNSON Due July 31
Assignment: Section 23: 23.1(d), 23.7, 23.8 Section 24: 24.1, 24.9, 24.17 Section 23 23.1 Find the radius of convergence and determine the exact interval of convergence. n3 (d) xn 3n