I'll discuss algebraic and geometric techniques for proving that categorifications of Kac-Moody and Heisenberg algebras have the "expected size" (in particular, that their Grothendieck groups really are Kac-Moody and Heisenberg algebras).  We'll emphasize the technique of using power series to simplify computations in these categories.  These talks will be based in part on joint work with Jon Brundan and Alistair Savage.