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Monotonic series for fractions near π and their convergents

Published online by Cambridge University Press:  22 June 2022

Stephen K. Lucas  and
Amrik Singh Nimbran
Stephen K. Lucas
Affiliation:
Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807, USA e-mail: lucassk@jmu.edu
Amrik Singh Nimbran
Affiliation:
B3/304, Palm Grove Heights, Ardee City, Gurugram, Haryana, India 122003 e-mail: amrikn622@gmail.com
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We describe various methods to derive monotonic infinite series for fractions near π and obtain a variety of series for the special case of its convergents. These series immediately show that π is clearly different from these fractions, replicating with series the results in Dalzell [1, 2] and Lucas that used integrals with non-negative integrands to represent the gaps between π and fractions.

Type
Articles
Information
The Mathematical Gazette , Volume 106 , Issue 566 , July 2022 , pp. 300 - 309
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/) which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Authors, 2022. Published by Cambridge University Press on behalf of The Mathematical Association

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