Plana's Summation Formula for ∑ ∞ m = 1,3, � m -2 sin(m α), m -3 cos(mα), m -2 A m , m -3 A m

Abstract

The four series oo oo ^ sin(2A: + 1)a/(2A: + 1)2 , J^ cos(2A: + 1)a/{2k + 1)3, 0 0 oo oo YgAlk+Xl(2k + \)1, and ^A2k+i/(2k + ¡)3 0 0 are very slowly convergent for 0 < a < 71 and as A-» 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.

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

“…Let us recall that considerable effort has been invested in the development of efficient numerical methods for the evaluation of the series (60) - (62) [55][56][57][58][59]. These alternative methods make intensive use of special properties of the series.…”

On the numerical evaluation of Legendre’s chi-function

“…In §3 the expansions are specialized for the cases n -2 and n = 3, and are then used in the numerical evaluation of the series (1.1) and (1.2). The present approach is shown to be more effective than the procedure in [2] based on Plana's summation formula. Yet another computational procedure has been developed by Gautschi [4].…”

“…The numerical evaluation of these slowly convergent series was recently discussed by Dempsey, Liu, and Dempsey [2]. These authors used Plana's summation formula (PSF) along with Romberg integration to improve the convergence of the series under consideration.…”

“…Recently, Dempsey, Liu, and Dempsey [2] presented a procedure for the numerical evaluation of the series which arise in the mathematical analysis of unilateral plate contact problems [6,Appendix B]. These series are slowly convergent, particularly if A is close to 1.…”

“…These series are slowly convergent, particularly if A is close to 1. To improve the convergence, the procedure in [2] uses Plana's summation formula, which in turn requires the computation of several definite integrals by Romberg integration. In this paper we develop a computational procedure which appears to be considerably simpler.…”