ON THE DERIVATION OF ITERATED SEQUENCE TRANSFORMATIONS FOR THE ACCELERATION OF CONVERGENCE AND THE SUMMATION OF DIVERGENT SERIES

Simple explicit expressions for named sequence transformations such as Wynn's rho algorithm (Proc. Cambridge Philos. Soc. 52 (1956) 663) and Drummond's sequence transformation (Bull. Aust. Math. Soc. 6 (1972) 69), which is used in connection with remainder estimates introduced by Levin (Int. J. Comput. Math. B 3 (1973) 371), are constructed. The elementary sequence transformations obtained in this way can be iterated leading to new nonlinear sequence transformations. Since the elementary sequence transformations, which are iterated, normally depend on n explicitly and not only implicitly via the sequence elements s(n), s(n)+1,...,s(n)+l, on which they act, there is a nonuniqueness problem which means that more than a single admissible iteration can usually be constructed. Numerical examples show that different admissible iterations of the same elementary sequence transformation may behave quite differently in convergence acceleration and summation problems.