On the highly accurate summation of certain series occurring in plate contact problems

Abstract:

The infinite series $R_p = \sum _{k=1}^\infty {(2 k - 1)}^{- p} x^{2 k - 1}$, $0 <1-x\ll 1$, $p = 2$ or $3$, and the related series \begin{equation*} \begin {split} C(x,b,2) &=\sum _{k=1}^\infty {(2k-1)}^{-2} \cosh (2k-1)x/\cosh (2k-1)b,\quad 0 <1-x/b \ll 1,\\ S(x,b,3)&=\sum _{k=1}^\infty {(2k-1)}^{-3} \sinh (2k-1)x/\cosh (2k-1)b, \end{split} \end{equation*} are of interest in problems concerning contact between plates and unilateral supports. This article will re-examine a previously published result of Baratella and Gabutti for $R_p$, and will present new, rapidly convergent, series for $C(x,b,2)$ and $S(x,b,3).$