In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students' activities, seeking to illustrate how the available tools mediated students' understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as "representational moderation".