### A challenging test for convergence accelerators: summation of a series with a special sign pattern.

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Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of...

We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $\pi ={\sum}_{n=0}^{\infty}((5n+3)n!\left(2n\right)!)/({2}^{n-1}(3n+2)!)$ with convergence as $13.{5}^{-n}$, in much the same way as the Euler transformation gives $\pi ={\sum}_{n=0}^{\infty}({2}^{n+1}n!n!)/(2n+1)!$ with convergence as ${2}^{-n}$. Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in the Appendix....