1. Prove that Vk € N, 1k + 2k + ... + nk € © (n²+1). 2. Suppose that the functions f1, f2, 91, 92: N→R20 are such that f₁ = O(g₁) and f₂ = O(92). Prove that (fi + ƒ2) € (max{91, 92}). Here (f1f2)(n) = f1(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}.

1. Prove that Vk € N, 1k + 2k + ... + nk € © (n²+1). 2. Suppose that the functions f1, f2, 91, 92: N→R20 are such that f₁ = O(g₁) and f₂ = O(92). Prove that (fi + ƒ2) € (max{91, 92}). Here (f1f2)(n) = f1(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I'd appreciate your assistance with these questions as I'm finding them challenging to comprehend. Could you please guide me step-by-step through the process, preferably by illustrating each step on paper? Please dont type up the solutions

1. Prove that
Vk € N, 1k + 2k + ... + nk € © (n²+1).
2. Suppose that the functions f1, f2, 91, 92: N→R20 are such that f₁ = O(g₁) and f₂ = O(92).
Prove that (fi + ƒ2) € (max{91, 92}).
Here (f1f2)(n) = f1(n) + f2(n) and max{91, 92}(n) = max{91(n), 92(n)}.

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