Let k be a positive integer. Show that 1k + 2k + … + nk is O(nk+1).
Step 1 of 3
Solution:Step-1: In this problem we need to show that is Note: Let us consider f and g are functions from the set of integers to the set of real numbers. The estimate value can be said that f(x) is O(g(x)) if there are constants C and k such that , where C > 0 and x> k.The constants C and k are called the witnesses to the relationship. The definition of f(x) is O(g(x)) says that f(x) grows slower than some fixed multiple of g(x) as x grows without bound.Step-2: Consider , , where k be any positive integer. , since 1 to n is increasing value(that is n> 1). , since Therefore , By using the above note , it is clear that C = 1, k = 1 and .Therefore , C = 1 and k = 1 are witnesses for .
Textbook: Discrete Mathematics and Its Applications
Author: Kenneth Rosen
The full step-by-step solution to problem: 18E from chapter: 3.2 was answered by , our top Math solution expert on 06/21/17, 07:45AM. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. The answer to “Let k be a positive integer. Show that 1k + 2k + … + nk is O(nk+1).” is broken down into a number of easy to follow steps, and 17 words. Since the solution to 18E from 3.2 chapter was answered, more than 400 students have viewed the full step-by-step answer. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This full solution covers the following key subjects: Integer, let, Positive, show. This expansive textbook survival guide covers 101 chapters, and 4221 solutions.