Exploring aspects of student facility with Taylor series in the context of the electric dipole

Physics students are typically first exposed to Taylor series expansions in their calculus courses, where the focus is often on mathematical procedures rather than on applying series to physical situations. Consequently, even students with strong mathematical backgrounds often struggle to recognize when and how to use Taylor series in physical contexts, particularly in middle- and upper-division physics courses. In this study, we investigated sophomore-level students' facility with Taylor series expansions in a physics context using one-on-one activity-based interviews centered around the electric dipole, for the purpose of characterization of common difficulties at this level. Our analysis reveals that while many students can perform the mechanics of a Taylor expansion, they frequently fail to identify common physics cues that signal when such an expansion is necessary. Additionally, we find that students often struggle to determine an appropriate expansion point and to understand its physical significance. We discuss several recurring difficulties and conceptual gaps that emerged in the interviews and consider the implications for instruction, with suggestions on how physics educators can aid students in supporting their procedural knowledge from mathematics with meaningful application in physics.