Taylor series is a topic briefly covered in most university calculus sequences. In many cases it constitutes only one or two sections of a calculus textbook. With this limited exposure, what do calculus students really understand about the convergence of Taylor series? Do they think of Taylor series convergence as a sequence of converging polynomials? Do they think of convergence as a remainder going to zero? Do they think the Taylor series for sine really "equals" sine, or is it merely a good estimation for sine? Furthermore, how might experts respond to these questions?

This study reported qualitative research methods which utilized multiple phases of data collection consisting of questionnaires and interviews from expert and novice (undergraduate student) participant groups. In addition, this study utilized multiple layers of analysis incorporating methods such as Strauss and Corbin's open coding and Sfard's discourse analysis. Using Tall and Vinner's notion of concept images, I analyzed and described the different ways in which both experts and novices conceptualized the convergence of Taylor series. In so doing, commonalities and differences amongst the expert and novice participant groups emerged.

The main result from this study was found in the descriptions of thirteen different concept images that experts and novices employed concerning the convergence of Taylor series. Some of these images were used more than others by the different participant groups, and some images appeared to date back to the early years of calculus. Even though both groups employed a variety of images, on an individual level, experts were more prone to use a wider range of images that they efficiently and effectively employed as different situations prompted. The most notable difference between experts and novices was found in their graphical images of Taylor series convergence. Experts demonstrated little to no difficulties interpreting graphs of Taylor series, but the vast majority of novices were unable to correctly produce graphs related to Taylor series convergence. In several cases, novices appeared to be incorrectly applying previous knowledge of graphical properties of translating functions in an attempt to build their conceptions of Taylor polynomial graphs. This finding has implications for future research into the effects of graphical images, both dynamic and static, on student conceptions of the convergence of Taylor series.