On certain slowly convergent series occurring in plate contact problems

Abstract:

A simple computational procedure is developed for accurately summing series of the form $\Sigma _{k = 0}^\infty {(2k + 1)^{ - p}}{z^{2k + 1}}$, where z is complex with $|z| \leq 1$ and $p = 2$ or 3, as well as series of the type \[ \sum \limits _{k = 0}^\infty {{{(2k + 1)}^{ - p}}\cosh (2k + 1)x/\cosh (2k + 1)b} \] and \[ \sum \limits _{k = 0}^\infty {{{(2k + 1)}^{ - p}}\sinh (2k + 1)x/\cosh (2k + 1)b} \], where $0 \leq x \leq b$, $p = 2$ or 3. The procedures are particularly useful in cases where the series converge slowly. Numerical experiments illustrate the effectiveness of the procedures.