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).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please help me with these question. I am having trouble understanding what to do. Please show all your work. Please do every question. I am study for my exam. And i need to know if I got the correct answer. And shown the correct steps just do questions 3 & 4
Course: Discrete mathematics for CS
Thank you
Transcribed Image Text: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).
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