Answered: PLEASE HELP ME. kindly show all your…

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter11: Nonlinear Programming

Section11.3: Convex And Concave Functions

Problem 13P

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PLEASE HELP ME. kindly show all your work

1. Prove that
∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1).

2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).
Prove that (f1 + f2) ∈ Θ(max{g1, g2}).

Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}.

3. Let n ∈ N \ {0}. Describe the largest set of values n for which you think 2n < 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 log2 n).

Thank you. But please show all work and all steps

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