Summing a common type of slowly convergent series of positive terms

J. E. Drummond

doi:10.1017/S0334270000001284

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If the terms of a series behave like n−k where k is an exactly known constant, a formula using two terms transforms the series into a series of terms like n−k −2 provided k ≠ 1. The multiple use of this transformation is demonstrated in summing three series.

References

[1]Lubkin, S., ‘A method of summing infinite series’, J. Res. Nat. Bur. Stand. 48 (1952), 228–254CrossRef Google Scholar

[2]Wynn, P.. ‘On a procrustean technique for the numerical transformation of slowly convergent sequences and series’, Proc. Cambridge Phil. Soc. 52 (1956), 663–671CrossRef Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Barber, Michael N. and Hamer, C. J. 1982. Extrapolation of sequences using a generalized epsilon-algorithm. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 23, Issue. 3, p. 229.

Drummond, J.E. 1984. Convergence speeding, convergence and summability. Journal of Computational and Applied Mathematics, Vol. 11, Issue. 2, p. 145.

Brezinski, Claude 1985. Convergence acceleration methods: The past decade. Journal of Computational and Applied Mathematics, Vol. 12-13, Issue. , p. 19.

Weniger, Ernst Joachim 1989. Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports, Vol. 10, Issue. 5-6, p. 189.

Osada, Naoki 1990. A Convergence Acceleration Method for Some Logarithmically Convergent Sequences. SIAM Journal on Numerical Analysis, Vol. 27, Issue. 1, p. 178.

Weniger, Ernst Joachim 1991. On the derivation of iterated sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Communications, Vol. 64, Issue. 1, p. 19.

1991. Extrapolation Methods Theory and Practice. Vol. 2, Issue. , p. 413.

Sablonniere, Paul 1991. Comparison of four algorithms accelerating the convergence of a subset of logarithmic fixed point sequences. Numerical Algorithms, Vol. 1, Issue. 2, p. 177.

Sablonnière, Paul 1992. Asymptotic behaviour of iterated modified 401-1401-1401-1401-2401-2401-2transforms on some slowly convergent sequences. Numerical Algorithms, Vol. 3, Issue. 1, p. 401.

Cioslowski, Jerzy and Weniger, Ernst Joachim 1993. Bulk properties from finite cluster calculations. VIII. Benchmark calculations of the efficiency of extrapolation methods for the HF and MP2 energies of polyacenes. Journal of Computational Chemistry, Vol. 14, Issue. 12, p. 1468.

Sablonnière, Paul 1995. Comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences. Journal of Computational and Applied Mathematics, Vol. 62, Issue. 1, p. 103.

Homeier, Herbert H.H. 1996. Analytical and numerical studies of the convergence behavior of the j transformation. Journal of Computational and Applied Mathematics, Vol. 69, Issue. 1, p. 81.

Homeier, H.H.H. 1999. Convergence acceleration of logarithmically convergent series avoiding summation. Applied Mathematics Letters, Vol. 12, Issue. 3, p. 29.

Homeier, H.H.H. 1999. Transforming logarithmic to linear convergence by interpolation. Applied Mathematics Letters, Vol. 12, Issue. 2, p. 13.

Weniger, Ernst Joachim 2000. Prediction properties of Aitken's iterated Δ2 process, of Wynn's epsilon algorithm, and of Brezinski's iterated theta algorithm. Journal of Computational and Applied Mathematics, Vol. 122, Issue. 1-2, p. 329.

Weniger, E.J and Kirtman, B 2003. Extrapolation methods for improving the convergence of oligomer calculations to the infinite chain limit of quasi-one-dimensional stereoregular polymers. Computers & Mathematics with Applications, Vol. 45, Issue. 1-3, p. 189.

Caliceti, E. Meyer-Hermann, M. Ribeca, P. Surzhykov, A. and Jentschura, U.D. 2007. From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions. Physics Reports, Vol. 446, Issue. 1-3, p. 1.

Paszkowski, Stefan 2011. Untypical methods of convergence acceleration. Numerical Algorithms, Vol. 56, Issue. 2, p. 185.

Bakulin, V. N. and Inflianskas, V. 2016. Addition to the Aitken method for the extrapolation of the limit of slowly convergent sequences. Computational Mathematics and Mathematical Physics, Vol. 56, Issue. 2, p. 191.

Chang, Xiang-Ke He, Yi Hu, Xing-Biao Sun, Jian-Qing and Weniger, Ernst Joachim 2020. Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order. Numerical Algorithms, Vol. 83, Issue. 2, p. 593.

Download full list

Article contents

Summing a common type of slowly convergent series of positive terms

Abstract

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests