1. Prove that Vk Є N, 1k+2k + ·+nk € © (nk+1). 2. Suppose that the functions f₁, f2, 91, 92 : N → R≥º are such that ƒ1 € ☹(91) and ƒ2 € ☹(92). Prove that (fi + ƒ2) € ©(max{91, 92}). Here (f1f2)(n) = fi(n) + ƒ₂(n) and max{91, 92}(n) = max{91(n), 92(n)}.

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Consider the function f(x) = - 3x/, which one of the following is true? O a) f is decreasing when I E (-∞, 0) and increasing when x E (0, 00). O b) None of these O c) f is increasing when x E (-00, 0) and decreasing when xE (0, 0). d) f is decreasing for all x E R. f is increasing for all ER.

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Let fi, f2,..., fkr 8ı, 82,..., g& be asymptotically nonnegative functions. Assume that fi(n) = O(g;(n)) for each i = 1,2,. ,k, prove that k k 1. ) fi(n) = 0(max{g;(n)} \1gisk 2. || f(n) = 0 8i(n) i=1 i=1 i=1

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