We'll begin by discussing the history of certain problems in Additive Number Theory. Several problems in Additive Number Theory ask how many ways we can represent the elements of a set A as a sum of s elements of the set B, with the two main examples being Waring's Problem and Goldbach's Conjecture. The Hardy-Littlewood Circle Method is the main tool for attacking these problems and often leads to asymptotic formulas for the number of representations. We'll introduce a variant of the Circle Method that simplifies the arguments involved in finding bounds for when the asymptotic formula holds in Waring's Problem.