Extended Aitken acceleration

The ordinary Aitken's ²-process uses three consecutive valuesx ₀,x ₁ andx ₂ of a linearly converging sequencex _i+1=(x _i ) to give an accelerated value with an error of order 2 in the error inx ₀. It is shown how the process may be extended in a simple way, very similar to the repeated Richardson'sh ²-extrapolation, to give an accelerated value with error of any orderk, using only thek+1 first order valuesx ₀,x ₁, ...,x _k .