Student: Isaac Zylstra Faculty Mentor: Professor John Ferdinands Summer 2014 Research Project #38: Sets of Selective Sums of Infinite Series "Sets of selective sums of infinite series" is an abstract mathematical concept. The main point is that different infinite series generate different kinds of sets of selective sums. The types of sets formed are not obvious and can be quite complex. Matching a type of selective sum to type of infinite series is a difficult task. These sets only show up briefly in a few scattered places in mathematical literature. So this summer, Professor Ferdinands, fellow student Andrew Hayes, and myself set out to discover whatever we could about these Sets of Selective Sums of Infinite Series. Our mathematical research was mainly about discovery. We have no idea if or when this research will be used in something else. In the first six or seven weeks, we focused on discovery, while in the past few weeks we have been writing the paper. In the discovery phase, most of the time was spent standing at a blackboard writing complex algebra (and fixing our algebra mistakes) or sitting, trying to come up with something new. We didn't prove anything formally, but haphazardly recorded the idea with the informal proof, so we could formally prove the theorem later. In the last few weeks, we've been writing the paper, finding our important results, and formally proving them. We came up with significant results, surpassing Professor Ferdinands' expectations, even excluding the problem that continues to elude us. Most of the results came in the form of either knowing the type of set of selective sums, or at least knowing what types were possible. We discovered a great deal about alternating series, that is, series whose terms alternate between positive and negative. We also discovered things about series where the ratio between terms approaches a certain number, and 2­dimensional series, where terms of the series can consist of different types of numbers. Finally we also looked at the different ways of representing individual selective sums in general series. This summer has been a great experience for myself personally. I have been seriously considering a career in mathematical research, a career requiring graduate school. This summer has been great in showing me what mathematical research is like. I think I'm leaning a bit farther away from mathematical research as a career now, but it's heartening to know that I'll either have a career in mathematical research, or something I find even more interesting.