ИСТИНА

Acceleration of slowly convergent series is one of the most traditional yet challenging topics of classical analysis. I will first speak about an applied problem -- an efficient evaluation of the quasi-periodic fundamental solution of 2-dimensional Laplace's operator, in a somewhat wider context of computational diffraction theory. As it turned out, an extremely powerful and useful modern method of series transformation, the Wilf-Zeilberger method, was essentially discovered by Andrei Markov back in 1890. In the second part of the talk I will outline this method, which is based on the finite-difference analog of Green's integral formula. Despite of (or due to) the very simple idea, the WZ technique yields fascinating results. Among its most remarkable applications are geometrically convergent series for the "Apery constant" $\zeta(3)$. We'll trace the history of world records about $\zeta(3)$ from 1880s to 2000s. In particular, a relevance of the accelerated series in Apery's proof of irrationality of $\zeta(3)$ will be discussed.