The Grothendieck groups of discrete cluster categories of Dynkin type A∞

In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type A∞ as introduced by Holm-Jørgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.