The sum of the series of reciprocals of the cubic polynomials with triple non-positive integer root

Authors

  • Radovan Potucek Department of Mathematics and Physics, Faculty of Military Technology, University of Defence, Brno, Czech Republic

DOI:

https://doi.org/10.23755/rm.v27i1.34

Keywords:

telescoping series, harmonic numbers, CAS Maple, Riemann zeta function.

Abstract

This contribution, which is a follow-up to author’s paper [1] dealing with the sums of the series of reciprocals of some quadratic polynomials, deals with the series of reciprocals of the cubic polynomials with triple non-positive integer root. Three formulas for the sum of this kind of series expressed by means of harmonic numbers are derived and presented, together with one approximate formula, and verified by several examples evaluated using the basic programming language of the computer algebra system Maple 16. This contribution can be an inspiration for teachers who are teaching the topic Infinite series or as a subject matter for work with talented students.

References

R. Potucek, The sums of the series of reciprocals of some quadratic polynomials. In: Proceedings of AFASES 2010, 12th International Conference ”Scientific Research and Education in the Air Force”. (CD-ROM). Brasov, Romania, 2010, 1206-1209. ISBN 978-973-8415-76-8.

Wikipedia contributors, Harmonic number. Wikipedia, The Free Encyclopedia, [online], [cit.2015-06-25]. Available from: https://en.wikipedia.org/wiki/Harmonic number.

E.W. Weisstein, Harmonic Number. From MathWorld – A Wolfram Web Resource, [online], [cit.2015-06-15]. Available from: http://mathworld.wolfram.com/HarmonicNumber.html.

Mathematics Stack Exchange – A question and answer web-site for people studying math. [online], [cit.2015-06-15]. Available from: http://math.stackexchange.com/questions/361386/is-there-a-formula-for-sum-n-1k-frac1n3?rq=1.

Downloads

Published

2014-12-14

Issue

Vol. 27 (2014)

Section

Volume 27-2014

License

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:

  • Free access for all users worldwide
  • Increased visibility and readership
  • Rapid publication
  • No spatial constraints
  • Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)

Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified. 
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.