Finding the generating function for the sequence of the sum of n^k

This thesis concerns the use of generating functions. Generating functions can be used to take recursive sequences and turn them into closed form functions. They can also be used to solve a variety of counting problems. We begin by investigating a simple recursive function a_(n+1) = a_(n) + n and developing techniques that allow for complicated equations to be solved more quickly. We investigate the possibility that there is a generating function that will work for all functions of the form a_(n+1) = a_(n) + n^k, which is the sequence of the sum of n^k, and explore if a closed form can be deduced for each generating function found.