Plana’s summation formula for $\sum ^ \infty _ {m=1,3,\cdots }m^ {-2}\sin (m\alpha ),m^ {-3}\cos (m\alpha ),m^ {-2}A^ m,m^ {-3}A^ m$
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- by Kevin M. Dempsey, Dajin Liu and John P. Dempsey PDF
- Math. Comp. 55 (1990), 693-703 Request permission
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.References
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M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Applied Mathematics Series No. 55, National Bureau of Standards, 1964.
- L. M. Delves and J. L. Mohamed, Computational methods for integral equations, Cambridge University Press, Cambridge, 1985. MR 837187, DOI 10.1017/CBO9780511569609 J. P. Dempsey, L. M. Keer, N. B. Patel, and M. L. Glasser, Contact between plates and unilateral supports, ASME J. Appl. Mech. 51 (1984), 324-328. J. P. Dempsey and Hui Li, Rectangular plates on unilateral edge supports: Part 1—Theory and numerical analysis, ASME J. Appl. Mech. 53 (1986), 146-150. —, Rectangular plates on unilateral edge supports: Part 2—Implementation; concentrated and uniform loading, ASME J. Appl. Mech. 53 (1986), 151-156.
- M. Lawrence Glasser, The summation of series, SIAM J. Math. Anal. 2 (1971), 595–600. MR 303156, DOI 10.1137/0502058 —, personal communication, Clarkson University, 1988. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1965. N. Grossman, personal communication, University of California at Los Angeles, 1988. E. R. Hansen, A table of series and products, Prentice-Hall, Englewood Cliffs, N.J., 1975.
- Peter Henrici, Elements of numerical analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0166900
- Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
- Peter Henrici, Essentials of numerical analysis with pocket calculator demonstrations, John Wiley & Sons, Inc., New York, 1982. MR 655251
- David Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math. 3 (1973), 371–388. MR 359261, DOI 10.1080/00207167308803075
- L. Lewin, Dilogarithms and associated functions, Macdonald, London, 1958. Foreword by J. C. P. Miller. MR 0105524 E. Lindelöf, Le calcul des résidus et ses applications à la théorie des fonctions, Gauthier-Villars, 1905; reprint, Chelsea, New York, 1947.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 55 (1990), 693-703
- MSC: Primary 65B10
- DOI: https://doi.org/10.1090/S0025-5718-1990-1035929-9
- MathSciNet review: 1035929