Untypical methods of convergence acceleration (Q625539)

The iterated Aitken \(\Delta^2\) method of convergence acceleration is modified for so-called quasi-geometrical series \(\sum_{j=0}^\infty\alpha_jx^j\) with the particular condition on the coefficients \(\alpha_j\) : \(\alpha_j=\sum_{l=0}^\infty c_lj^{\theta -l}\), \(c_0\neq 0\) with \(\theta\) real. Some examples are given where \(\alpha_j\) is a rational function of \(j\) and \(\theta\) an integer. For this kind of series two modified Aitken methods considerably accelerate the convergence with respect to the classical formulas. Another method, so-called ``irrational method'' is proposed and in some cases compared with the previous methods. In many cases the partial sums accelerated by this method are more accurate than analogous approximants evaluated by Aitken method. The modification of Levin and Weniger convergence acceleration methods by introduction of an auxiliary parameter improves the results.