The Acceleration Processes of Aitken and Euler

Abstract

We have seen that there is a great variation in the speed of convergence of series or sequences. Some are too slowly convergent to be summed directly on the fastest computer. This is the case of the logarithmic series

$$\log 2 = 1 - {\textstyle{1 \over 2}} + {\textstyle{1 \over 3}} - {\textstyle{1 \over 4}} + \ldots $$

in which a crude estimate for the remainder after _n terms is n⁻ so that 2 × 10⁶ terms would be needed to get a value of log 2 correct to 6D.