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/
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
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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
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
NAME: _____ Student ID:
a MA238 Winter 2009 - Quiz 2a
Total marks is . Answer all questions in the spaces provided. Calculators are not permitted.
3
wmarks] 1. Given the following adjacency matrix, draw the corresponding graph.
A
b
c
OHOO§
OOvIOU
OOo
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.
CD
[ marks] 1. (a) What is the denition of an Euler path?
A A IEU-lc, [pa-H it a path #4410 3'19
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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
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
Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 12 31.2 Find the Taylor series for sinh x = (ex ex )/2 and cosh x = (ex + ex )/2. Solution. The result x2n1 x2n sinh x = , cosh x = (2n 1)! (2n)! n1 n0 follows easily from either the
Math 3210-3 HW 16
Solutions
Properties of Continuous Functions
1. Show that 2x = 3x for some x (0, 1). Proof: First I claim that f (x) = 2x is a continuous function on [0, 1]. Let's assume ex and ln x are continuous functions on R. Then consider y = ln 2x
2.4. Suppose for the sake of contradiction that x = (5 3)1/3 represents a rational number. Then we can write x as p where p and q q are integers with no common factors. Also, we have that x satises (x3 5)2 = 3 x6 10x3 + 22 = 0 By the rational zeros theor
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