In analogy with the chromatic convergence theorem of Hopkins and Ravenel, Waldhausen conjectured that the natural map K(S) \to \holim_n K(L_n S) is a weak equivalence. In this talk, I will describe joint work with Mandell and Yuan that establishes that this map is the inclusion of a wedge summand. Along the way we give an explicit calculation of holim_n L_n K(S), the chromatic completion of K(S). Speaker(s): Andrew Blumberg (Columbia University)