 Cubic convergence

An iteration has at least cubic convergence when its errors eventually satisfy the displayed inequality with fixed $C>0$. Exact order of convergence three requires a positive finite limit of $\Vert e_{k+1}\Vert /\Vert e_k{\Vert }^3$. If $C^{1/2}\Vert e_N\Vert <1$, setting $d_k=C^{1/2}\Vert e_k\Vert$ gives $d_{k+1}\le d_k^3$, hence $d_{N+j}\le d_N^{3^j}$.