I currently work as a substitute teacher at a local high school and the topic in one of the classes is sequences, series and limits. Because I always disliked learning about a topic without having some sense of what problem it addresses or what new possibilities it opens, I wanted to present a motivating example in the first lesson. What I had in mind was an application of sequences, series or limits that is easy enough to grasp without already knowing the topic but still reliant on its ideas to motivate its study.
To clarify some of the terms I am using: By sequences, series and limits I mean the basic theory of sequences (explicit formulas, recursive formulas, arithmetic sequence, geometric sequence, divergent and convergent sequences), the simple series (arithmetic series, geometric series, divergence of the harmonic series) and the definition of convergence using an epsilon bound.
After doing some research on the web I went with Zenos paradox of Achilles and the Tortoise and after presenting it, letting the students discuss it among themselves and arguing Zenos position, I showed them how the paradox can be resolved with an infinite series. In the last part however, I had to simply state that this infinite summation yields a finite value and that proving this fact and similar things would be the content of the following lectures. Afterwards I felt like I had presented a solution that wasn't really understandable and that left people rather disappointed than hungry for more. So my question is if any of you know motivating examples for the study of sequences, series and limits to get high school students interested in the topic? Ideally I would like these examples to highlight the importance of dealing with infinities. I know there are plenty of cool problems involving sequences and series that connect to geometry, physics or finance ( for example here: Examples of arithmetic and geometric sequences and series in daily life ) but I haven't found ones that met my criteria of demonstrating the need for sequences, series and dealing with infinity in an accessible way.