### The Sum of the Series of Reciprocals of the Quadratic Polynomials with Complex Conjugate Roots

#### Abstract

This contribution is a follow-up to author’s papers [1], [2], [3], [4], [5], [6], [7], and in particular [8] dealing with the sums of the series of reciprocals of quadratic polynomials with different positive integer roots, with double non-positive integer root, with different negative integer roots, with double positive integer root, with one negative and one positive integer root, with purely imaginary conjugate roots, with integer roots, and with the sum of the finite series of reciprocals of the quadratic polynomials with integer
purely imaginary conjugate roots respectively. We deal with the sum of the series of reciprocals of the quadratic polynomials with complex conjugate roots, derive the formula for the sum of these series and verify it by some examples evaluated using the basic programming language of the computer algebra system Maple 16. This contribution can be an inspiration for teachers of mathematics who are teaching the topic Infinite series or as a subject matter for work with talented students.

#### Keywords

sum of the series; harmonic number; imaginary conjugate roots; hyperbolic cotangent; computer algebra system Maple

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#### References

R. Potucek, The sum of the series of reciprocals of the quadratic polynomial with different positive integer roots. In: Mathematics, Information Technologies and Applied Sciences (MITAV 2016). University of Defence, Brno, 2016, p. 32-43. ISBN 978-80-7231-464-5.

R. Potucek, The sum of the series of reciprocals of the quadratic polynomial with different negative integer roots. In: Ratio Mathematica - Journal of Foundations and Applications of Mathematics, 2016, vol. 30, no. 1/2016, p. 59-66. ISSN 1592-7415.

R. Potucek, The sum of the series of reciprocals of the quadratic polynomials with double non-positive integer root. In: Proceedings of the 15th Conference on Applied Mathematics APLIMAT 2016. Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 2016, p. 919-925. ISBN 978-802274531-4.

R. Potucek, The sum of the series of reciprocals of the quadratic polynomials with one negative and one positive integer root. In: Proceedings of the 16th Conference on Applied Mathematics APLIMAT 2017. Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 2017, p. 1252-1253. ISBN 978-80-227-4650-2.

R. Potucek, The sum of the series of reciprocals of the quadratic polynomials with double positive integer root. In: MERAA – Mathematics in Education, Research and Applications, 2016, vol. 2, no. 1. Slovak University of Agriculture in Nitra, 2016, p. 15-21. ISSN 2453-6881.

R. Potucek, The sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots. In: MERAA – Mathematics in Education, Research and Applications 2017, vol. 3, no. 1. Slovak University of Agriculture in Nitra, 2017, p. 17-23. ISSN 2453-6881, [online], [cit.2018-04-30]. Available from: http://dx.doi.org/10.15414/meraa.2017.03.01.17-23.

R. Potucek, The sum of the series of reciprocals of the quadratic polynomials with integer roots. In: Proceedings of the 17th Conference on Applied Mathematics APLIMAT 2018. Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 2018, p. 853-860, ISBN 978-80-227-4765-3.

R. Potucek, The sum of the finite series of reciprocals of the quadratic polynomial with integer purely imaginary conjugate roots. In: Mathematics, Information Technologies and Applied Sciences (MITAV 2018). University of Defence in Brno, Brno, 2018, p. 101-107, ISBN 978-80-7582-040-2.

Wikipedia contributors, Harmonic number. Wikipedia, The Free Encyclopedia, [online], [cit.2018-04-30]. Available from: https://en.wikipedia.org/wiki/Harmonic number.

DOI: http://dx.doi.org/10.23755/rm.v35i0.427

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