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Please help me with these question. SHow all you work. Thank you 1. Prove that VKE N, 1k+ 2k + · · · + nk € ©(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 = O(g1) and f2 = ☹(g2). Prove that (f1+f2) = O(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}

Please help me with these question. SHow all you work. Thank you 1. Prove that VKE N, 1k+ 2k + · · · + nk € ©(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 = O(g1) and f2 = ☹(g2). Prove that (f1+f2) = O(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}

Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter2: Basic Linear Algebra
Section: Chapter Questions
Problem 15RP
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Please help me with these question. Show all you work. Thank you

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)}

Please help me with these question. SHow all you work. Thank you 1. Prove that VKE N, 1k+ 2k + · · · + nk € ©(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 = O(g1) and f2 = ☹(g2). Prove that (f1+f2) = O(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}
Transcribed Image Text:Please help me with these question. SHow all you work. Thank you 1. Prove that VKE N, 1k+ 2k + · · · + nk € ©(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 = O(g1) and f2 = ☹(g2). Prove that (f1+f2) = O(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}
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