1. Prove that Vk € N, 1k + 2k + . . . +nk € ©(nk+1). 2. Suppose that the functions f1, f2, 91, 92 : N → R≥0 are such that f₁ = O(91) and ƒ2 Є ☹(92). Prove that (f1 + ƒ2) € ☹(max{91,92}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}. 3. Let n Є N\{0}. Describe the largest set of values n for which you think 2″ < n!. Use induction to prove that your description is correct. Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! € O(n log n).

1. Prove that Vk € N, 1k + 2k + . . . +nk € ©(nk+1). 2. Suppose that the functions f1, f2, 91, 92 : N → R≥0 are such that f₁ = O(91) and ƒ2 Є ☹(92). Prove that (f1 + ƒ2) € ☹(max{91,92}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}. 3. Let n Є N\{0}. Describe the largest set of values n for which you think 2″ < n!. Use induction to prove that your description is correct. Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! € O(n log n).

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Similar questions

hello im new, i only get 3 free questions and im failing i wish there was a way to grade up but there isn't so please help me with what you can i beg you,.
I need help with just part c. Someone on here already answered it for me but I'm really confused as there are steps missing and I'm not sure what to put in the calculator. " The first quartile separates the lowest 25% of the observations from the higher 75% of the observations. The first quartile is denoted by Q1. The second quartile separates the lower 50% of the observations from the higher 50% of the observations. The second quartile is denoted by Q2.The second quartile is same as median. The third quartile separates the lowest 75% of the observations from the higher 25% of the observations. The third quartile is denoted by Q3. The third quartile can obtained as follows...." Third Quartile is 1 somehow, and I don't understand how it is.
Please answer math question number 4. **Fill in all blanks** This is a practice question for my son so it is not taken for acadamic purposes. Please only type the answer out, this will ensure a thumbed up answer. Other answer that are otherwise written in purpose will be given the opposite (thumbs down)
What are some of the instruments that can be used to conduct a research project? How does one select the correct one?
Hello. Please answer the attached Discrete Mathematics question correctly and follow all directions. Do not copy the answers already given on Chegg. I just want your own personal answer. *If you would like for me to give you a thumbs up then just answer the question by yourself and do not copy from outside sources. Thank you.
How would I answer this question
need answer and all the work
Please take a screenshot of all the steps ******************************** Q 1 (a) Enter the following data into PSPP : STUDENT NAME DEPARTMENT COURSE MARKS Tommy 1 Computer Networks 75 John 2 Software Engineering 87 Anabell 1 Programming 94 Rose 2 Information Technology 50 Sarah 2 Software Engineering 72 Value=1 represents “CS” Value=2 represents “IT” Perform the following on the above data: Using the Descriptive analysis calculate the Sum, Mean, Mode and Standard deviation for Marks Do a Frequency analysis on the variable “Department” and create a Pie chart for
Please clearly shows the steps. I'm not sure if my answer(1/3) is correct, so please just ignore it and calculate by yourself.
Can you help me please? What is the most interesting piece of history from Greek Mathematics. Why is it interesting to you? How is it important to mathematics as a whole?