These talks will be self-contained presentations of mathematical concepts and theorems. They are intended to be accessible to all mathematics students.
 
 

Why should the sum of the kth powers of the first n natural numbers be a polynomial function of n? Call this polynomial S_k(x). These polynomials have a number of properties that are elementary to prove but difficult to relate to the sums that define them. For example, why should S_k (-1)= 0? Why should the fourth highest power of x always vanish? Why and how is S_k+1(x) determined by the antiderivative of S_k(x)? Why should the number x = -1/2 be the "centre of the universe" for these polynomials? Prerequisites for this talk: the binomial formula and a little calculus.