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In this class, we'll explore some interesting and beautiful things you can do with summations, illuminating some interesting techniques and surprising mathematical connections. Planned material: exploring reciprocals of triangular numbers, examining the familiar and beloved harmonic series, and visiting Euler's solution of the Basel's problem. We'll also examine how using telescoping series gives us insight into summing consecutive integer powers, as well as a geometric interpretation of the result for degree 3. As time allows, we'll also spend some time working through some problems together that illustrate some of the ideas presented. The material briefly touches on non-elementary ideas, such as Taylor series and results we will not prove, but familiarity with these concepts is not at all necessary for understanding the material.

Set theory provides a way to formalize all of mathematics using a single language, but different axiomatizations can result in different mathematical results. This fact, interesting in itself, may not be surprising to those who are familiar with the idea of set theory as foundations for mathematics, but in this class we present a graph which is surprisingly easy to define, and show that it behaves very differently when we make a simple adjustment to what we want to do with the axiom of choice.