This article deals with sums of infinite geometric series. We focus on the understanding of the notion by pupils at secondary school through generic and universal models. In the first part we survey this notion in the Czech and Slovak curriculum. We describe the process of gaining knowledge as a sequence of five stages. In the second part we show one possible approach how to introduce the notion "sum of the infinite geometric series" through this process. We illustrate this on some examples for pupils. At the end we formulate some pedagogical recommendation for teachers.