SOLUTIONS Math 104. Midterm Exam. 02.28.06 Problem 1. Formulate the denition of a Cauchy sequence (full sentences, please). A sequence (xn ) (of real numbers) is called a Cauchy sequence if for every > 0 there exists a natural number N such that for all m
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 312, Intro. to Real Analysis: Homework #7 Solutions
Stephen G. Simpson Wednesday, April 29, 2009
The assignment consists of Exercises 20.1, 20.18, 23.1, 23.4, 23.6, 24.1, 24.2, 24.6, 24.14, 24.17, 25.3, 25.6, 26.2, 26.6, 26.7 in the Ross textbook. Ea
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 312, Intro. to Real Analysis: Homework #4 Solutions
Stephen G. Simpson Monday, March 2, 2009
The assignment consists of Exercises 10.6, 10.8, 10.10, 11.2, 11.3, 11.6, 11.10, 12.2, 12.12 in the Ross textbook. Each problem counts 10 points. 10.6. (a) L
Math 312, Intro. to Real Analysis: Homework #1 Solutions
Stephen G. Simpson Wednesday, January 21, 2009
The assignment consists of Exercises 1.4, 1.7, 1.11, 2.4, 3.4, 3.7 in the Ross textbook. Each problem counts 10 points. 1.4 (a) 1 = 1; 1 + 3 = 4; 1 + 3
Math 104, Solution to Quiz 12
Instructor: Guoliang Wu August 10, 2009
1. (15 points) Prove that | sin x sin y | |x y |, for all x, y R. Proof. If x = y , then obviously | sin x sin y | = |x y | = 0. If x = y , by the mean value theorem , there exists x0 b
Math 104, Solution to Quiz 2
Instructor: Guoliang Wu June 29, 2009
1. (15 pts) Use Triangle Inequality to prove that for any four real numbers x1 , x2 , x3 , x4 , we have |x1 x2 | + |x2 x3 | + |x3 x4 | + |x4 x1 | |x1 x3 | + |x2 x4 |. Hint: Consider the fo
Math 104, Solution to Quiz 8
Instructor: Guoliang Wu July 21, 2009
1. (15 points) Prove that the following equation has at least one real solution in (0,1): 3 2 ex = x + . 2 Proof. Let f (x) = ex (x + 3 ). Then f is continuous on [0, 1]. 2 3 1 = < 0; 2 2
Math 104, Solution to Quiz 3
Instructor: Guoliang Wu July 1, 2009
1. (5 pts3) Determine the following limits without proof. (Just write down the answer.) (a) lim n1/n = 1.
n
Solution: For a formal proof, see textbook, Example 9.7. (b) lim ( n2 + 2 n) = 0.
Math 104, Solution to Quiz 5
Instructor: Guoliang Wu July 14, 2009
1. (9 points) Fill in the following table. Sequences Set of subsequential limits lim sup lim inf (1)n n2 cfw_,
n sin 3 cfw_0, 3 , 3/2 /2 3 /2 3/2 (1)n n
cfw_0 0 0
2. (6 points) Are the f
Math 312, Intro. to Real Analysis: Final Exam: Solutions
Stephen G. Simpson Friday, May 8, 2009
1. True or false (3 points each). (a) For all sequences of real numbers (sn ) we have lim inf sn lim sup sn . True. (b) Every bounded sequence of real numbers
Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1 1.1. Prove that 12 + 22 + + n2 = 6 n(n + 1)(2n + 1) for all n N. Put f (n) = n(n + 1)(2n + 1)/6. Then f (1) = 1, i.e the theorem holds true for n = 1. To prove the theorem, it su
SOLUTIONS TO HOMEWORK 1 AND 2
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Homework Assignment 3
Solutions
1. Let A, B IRn . Prove that (intA) (intB) = int(A B). Is the statement true if
intersection is replaced by union? Explain.
Solution. To show that two