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...

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