Let k be a positive integer. Show that 1k + 2k + … + nk is O(nk+1).

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 .