Asymptotic expansions in $n^{-1}$ for percolation critical values on the $n$-cube and ${\mathbb{Z}}^n$

We use the lace expansion to prove that the critical values for nearest-neighbour bond percolation on the $n$-cube $\{0,1{\}}^n$ and on ${\mathbb{Z}}^n$ have asymptotic expansions, with rational coefficients, to all orders in powers of $n^{-1}$.