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Inverse tangent series via telescoping sums

Published online by Cambridge University Press:  12 November 2024

Russell A. Gordon
Russell A. Gordon*
Affiliation:
Department of Mathematics and Statistics, Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362, USA e-mail: [email protected]
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When first learning about infinite series, students typically are shown some examples for which the partial sums can be simplified by taking advantage of telescoping sums. In this paper, we present many examples of such series, all involving the inverse tangent function and most of which involve the Fibonacci and Lucas numbers. Most of the series presented here have appeared in various papers (see the references), but the authors are usually working in an abstract setting which makes it difficult for students to follow the basic ideas. We seek to make these results accessible to a wider audience.

Type
Articles
Information
The Mathematical Gazette , Volume 108 , Issue 573 , November 2024 , pp. 419 - 429
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Copyright
© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

References

Boros, G. and Moll, V., Sums of arctangents and some formulas of Ramanujan, Scientia Series A: Mathematical Sciences 11 (2005) pp. 13–24.Google Scholar
Heaslet, M., Solution to Problem 3801 (posed by D. Lehmer), Amer. Math. Monthly 45 (1938) pp. 636-637.Google Scholar
Hoggatt, V. and Ruggles, I., A primer for the Fibonacci numbers–Part V, Fibonacci Quart. 2:1 (1964) pp. 46–51.CrossRefGoogle Scholar
Mahon, J. and Horadam, A., Inverse trigonometrical summation formulas involving Pell polynomials, Fibonacci Quart. 23 (1985) pp. 319–324.CrossRefGoogle Scholar
Guo, D. and Chu, W., Inverse tangent series involving Pell and Pell-Lucas polynomials, Math. Slovaca 72 (2022) pp. 869–884.CrossRefGoogle Scholar
Adegoke, K., Infinite arctangent sums involving Fibonacci and Lucas numbers, Notes on Number Theory and Discrete Mathematics 21 (2015) pp. 56–66.Google Scholar
Bragg, L., Arctangent sums, College Math. J. 32 (2001) pp. 255–257.CrossRefGoogle Scholar
Frontczak, R., Further results on arctangent sums with applications to generalized Fibonacci numbers, Notes on Number Theory and Discrete Mathematics 23 (2017) pp. 39–53.Google Scholar
Hoggatt, V. and Ruggles, I., A primer for the Fibonacci numbers–Part IV, Fibonacci Quart. 1:4 (1963) pp. 39–45.Google Scholar
Wu, R., Generalizations of Lehmer’s arctangent sum, IIME Journal 15 (2022) pp. 355–363.Google Scholar