Math 104, Solution to Homework 1
Instructor: Guoliang Wu June 19, 2009
Ross, K. A., Elementary Analysis: The theory of calculus: 1.2 Prove 3 + 11 + + (8n 5) = 4n2 n for all natural numbers n. Solution: (1) When n = 1, the assertion holds since 3 = 4 12 1.
Math 104, Solution to Quiz 10
Instructor: Guoliang Wu August 1, 2009
1. State the equivalent formulation of limits of the functions. (a) Suppose a, L R. lim f (x) = L if and only if:
xa
for each
> 0, there exists > 0 such that
0 < a x < implies |f (x) L|
Math 104, Solution to Quiz 9
Instructor: Guoliang Wu July 30, 2009
x2 +1 x 1 .
1. Consider the function f (x) =
(a) (7 pts) Is f uniformly continuous on I1 = [2, 3]? Solution: f is uniformly continuous on I1 = [2, 3]. Since both x2 + 1 and x 1 are continu
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 7
Instructor: Guoliang Wu July 18, 2009
1. Are the following series convergent or divergent? Please justify your answers.
(a) (7 points )
n=2
(1)n ln n
1 Solution: The sequence an = ln n is nonnegative, decreasing (since ln n is
Math 104, Solution to Quiz 6
Instructor: Guoliang Wu July 16, 2009
1. (15 points) Is the following series convergent? Justify your answer. arctan n . 2n + 1 Solution: For any n N, 0 arctan n /2. Hence 0 arctan n /2 n. n+1 2 2
/2 Since 2n is a geometric s
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 104, Solution to Quiz 4
Instructor: Guoliang Wu July 6, 2009
1. (a) (5 points) Suppose lim an = 2 and lim bn = 1. Find the
n n
following limit. (No proof required.) 3 2an b2 n =. n bn + 1 2 lim 2an b2 (2)(2) 12 3 n = =. n bn + 1 1+1 2 (b) (10 points)
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 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 1
Instructor: Guoliang Wu June 20, 2009
1. (15 pts) Use mathematical induction to prove that 1 + r + r2 + + rn = 1 rn+1 , 1r
1 r 2 1 r .
for any natural number n and any r = 1. Solution: (1) When n = 1, the identity holds since
Math 312, Intro. to Real Analysis: Midterm Exam #2
Stephen G. Simpson Friday, March 27, 2009
1. True or False (2 points each) (a) Every monotone sequence of real numbers is convergent. (b) Every sequence of real numbers has a lim sup and a lim inf. (c) Ev
Math 312, Intro. to Real Analysis: Midterm Exam #2 Solutions
Stephen G. Simpson Friday, March 27, 2009
1. True or False (2 points each) (a) Every monotone sequence of real numbers is convergent. False. (b) Every sequence of real numbers has a lim sup and
Math 312, Intro. to Real Analysis: Midterm Exam #1
Stephen G. Simpson Friday, February 13, 2009
1. True or False (3 points each) (a) Every ordered eld has the Archimedean property. (b) The ordered eld axioms imply |a b| |a| + |b| for all a, b. (d) For any
Math 312, Intro. to Real Analysis: Midterm Exam #1 Solutions
Stephen G. Simpson Friday, February 13, 2009
1. True or False (3 points each) (a) Every ordered eld has the Archimedean property. Answer: False. (b) The ordered eld axioms imply |a b| |a| + |b|
Math 312, Intro. to Real Analysis: Final Exam
Stephen G. Simpson Friday, May 8, 2009
There are problems worth a total of 150 points. Please be sure to attempt all problems. 1. True or false (3 points each). (a) For all sequences of real numbers (sn ) we h
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
Math 104, Midterm Examination 2 Solution
Instructor: Guoliang Wu
1. (10 points) Find the following limit and justify your answer.
n
lim
1 [(2n)!]1/n 2 n
Solution: Let sn = Since
(2n)! , then we need to nd the limit limn (sn )1/n . n 2n
n 2n sn+1 (2n + 2)!
MATH 104, SUMMER 2008, REVIEW SHEET FOR MIDTERM 2
To do well on this exam you should be able to do at least each of the following1: Section 11: Subsequences (1) Statement of the Bolzano-Weierstrass Theorem and its applications. (2) Dene subsequential limi
Math 104, Practice Midterm 1 Solution
Please try to solve these problems by yourselves rst. Please let me know if you nd any typos/mistakes. 1. Use limit theorems to evaluate the following limits. Justify each step. (a) lim 5n2 + sin n n n2 + 1 Solution:
Math 104, Practice Midterm 1
The rst midterm will cover all the following sections: 1 5 and 7 11 in Ross. Time: July 9, 2:30-4:00pm, in lecture room 3107 Etcheverry. It will be a closed-book, closed-notes exam and no calculator will be allowed. The follow
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