Plana’s summation formula for $\sum ^ \infty _ {m=1,3,\cdots }m^ {-2}\sin (m\alpha ),m^ {-3}\cos (m\alpha ),m^ {-2}A^ m,m^ {-3}A^ m$

Abstract:

The four series \[ \begin {array}{*{20}{c}} {\sum \limits _0^\infty {\sin (2k + 1)\alpha /{{(2k + 1)}^2},} \quad \sum \limits _0^\infty {\cos (2k + 1)\alpha /{{(2k + 1)}^3},} } \\ {\sum \limits _0^\infty {{A^{2k + 1}}/{{(2k + 1)}^2},\quad {\text {and}}\quad \sum \limits _0^\infty {{A^{2k + 1}}/{{(2k + 1)}^3}} } } \\ \end {array} \] are very slowly convergent for $0 \leq \alpha \leq \pi$ and as $A \to {1^ - }$. Direct summation involves thousands of terms to get the accuracy desired. Plana’s summation formula along with Romberg’s method of integration significantly and consistently improves the convergence and accuracy for the above series.