Reference no: EM131081421
Math 104: Homework 2-
1. Consider each of the following sets:
A = (0, ∞),
B = {1/m + 1/n|m, n ∈ N},
C = {x2 - x - 1 | x ∈ R},
D = [0, 1] ∪ [2, 3],
E = n=1∪∞[2n, 2n + 1],
F = n=1∩∞(1 - 1/n , 1 + 1/n).
For each set, determine it's minimum and maximum if they exist. In addition, determine each set's infimum and supremum, writing your answers in terms of infinity for unbounded sets. Detailed proofs are not required.
2. Let A and B be nonempty bounded subsets of R, and let M = {a · b | a ∈ A, b ∈ B}. Is sup M = (sup A) · (sup B)? Either prove, or provide a counterexample.
3. Find the limits of each of the following sequences, defined for n ∈ N:
(a) (3n/n+3)2,
(b) 1+2+...+n/n2,
(c) an-bn/an+bn for a > b > 0,
(d) n2/2n,
(e) √(n + 1) - √n.
Detailed proofs are not required, but you should justify your answers.
4. Let sn = n!/nn for all n ∈ N. Prove that sn → 0 as n → ∞.
5. Let (sn)n∈N be a sequence such that sn → s as n → ∞. Prove that if p : R → R is a polynomial function, then p(sn) → p(s). [Hint: Use the limit theorems for addition and multiplication of sequences.]
6. Let s1 = t for some t ∈ R, and define a sequence according to sn+1 = 1 + (sn/2) for n ∈ N. Prove that for all t, sn → 2 as n → ∞.
7. Optional for the enthusiasts. Let s1 = t for t ∈ R, and define sn+1 = sn/2(3 - s2n) for n ∈ N. For all values of t, determine and prove whether sn is convergent, divergent but bounded, or divergent and unbounded.
Time value of money and discounted cash flow analysis
: This week we are studying the time value of money and discounted cash flow (DCF) analysis. In order to use DCF one must make projections of future cash flows. The DFC analysis is therefore only as good as the quality of the projections. For this week..
|
What strategies can be used to increase ticket sales
: What strategies can be used to increase merchandise sales, from the standpoint of both the product selection and promotion during the concert?
|
Responses of individual americans
: The Great Depression plunged the nation into a profound crisis with staggering personal and national costs. How did Americans attempt to ease the impact of these circumstances? In your answer, discuss and compare the responses of individual Americ..
|
Writing a report about the expriment meter circuits
: Writing a report about the expriment meter circuits. Your report should include A discussion of the technique for measuring the meter internal resistance.
|
Determine and prove whether is convergent and divergent
: Math 104: Homework 2. Optional for the enthusiasts. Let s1 = t for t ∈ R, and define sn+1 = sn/2(3 - s2n) for n ∈ N. For all values of t, determine and prove whether sn is convergent, divergent but bounded, or divergent and unbounded
|
Reflect the mexican revolution
: Please refer to my Research Proposal as the paper needs to reflect the Mexican Revolution of 1910. Write a first draft of your research paper. The first draft should be 3,000-3,750 words (approximately 12-15 pages if the template is used correctly)..
|
What about raising the rent on the properties
: What about raising the rent on the properties?
|
Arrogance of wealth
: In 1914 President Woodrow Wilson denounced what material object as a symbol of "the arrogance of wealth"?
|
Credited with building the first cities
: Credited with building the first cities in Italy and introducing Greek culture to Rome, what people occupied much of Italy before being supplanted by the Romans in the third century B.C?
|