The Markov-WZ method. The authors introduce a generalization of the well-known WZ theory given by Wilf and Zeilberger using ideas which are due to Markov and were used for convergence-acceleration of infinite series already in 1890. It turns out that the two methods can be combined and give a new algorithmic procedure for convergence-acceleration. The authors introduce a Maple package Markov WZ which can be downloaded from the second author’s website and give some examples.