is a weak equivalence due to the chromatic convergence theorem of M. Hopkins and D. Ravenel [@rav2]. F. Waldhausen’s idea was to use his localization theorem to understand the algebraic K-theory of the fibers of the maps $$K(L_{E(n)}S_{(p)}) \lra K(L_{E(n-1)}S_{(p)})$$ and use an inductive argument to understand the whole tower. Unfortunately, Waldhausen’s argument was based on the validity of the telescope conjecture relating the finite localizations $L_{E(n)}^fS_{(p)}$ to $L_{E(n)}S_{(p)}$, which is now believed to be false for $n>1$. However, using a Thomason-Trobaugh type localization theorem in nonconnective K-theory, B. Antieau, T. Barthel, and D. Gepner provide an identification of the fiber that does not rely on the telescope conjecture [@1412.4041].

It is also natural to ask if the chromatic convergence theorem holds after applying algebraic K-theory. J. McClure and R. Staffeldt showed that after taking connective covers the tower converges; i.e. the map $$K(S_{(p)}) \lra \underset{\leftarrow}{\holim}\hspace{.05in} K(\tau_{\ge 0}L_{E(n)}S_{(p)} )$$ is a weak equivalence where we write $\tau_{\ge 0} L_{E(n)}S_{(p)}$ for the connective cover functor applied to the $E(n)$-local sphere [@MR1164148]. By convention, the first object in the tower is $K(\mathbb{Z}_{(p)})$ and the first map is the induced map in algebraic K-theory of the linearization map $\tau_{\ge 0}L_{E(1)}S_{(p)}\rightarrow \mathbb{Z}_{(p)}$. In fact, 