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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).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 52E
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Course: Discrete mathematics for CS

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just question 3 & 4

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).
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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