PreprintCC BY-SA 4.0arXiv
The method of telescoping continued fractions
Abstract
We give an approach to discover continued fractions for series of the form ∑k=0∞ εk/((x+k)s), where ε= ± 1. We find a continued fraction of the form a1/b1(x) fplus a2/b2(x) fplus fdots where ak are constants and bk(x) are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for 2∑k=1∞ ((−1)k+1)/(x+2k−1), 2∑k=0∞ 1/((x+2k+1)2), 2∑k=0∞ ((−1)k)/((x+2k+1)2), ∑k=1∞ 1/((x+k)3), and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of s, including s=5, 7, 9, 11. They do not follow as simple a pattern as Ramanujan's continued fractions.
Related
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BibTeX
@misc{bhatnagar2026method,
title = {{The method of telescoping continued fractions}},
author = {Gaurav Bhatnagar and Krishnan Rajkumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07092},
eprint = {2609.07092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07092},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The method of telescoping continued fractions
AU - Gaurav Bhatnagar
AU - Krishnan Rajkumar
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07092
UR - https://arxiv.org/abs/2609.07092
AB - We give an approach to discover continued fractions for series of the form ∑_k=0^∞ ε^k/((x+k)^s), where ε= ± 1. We find a continued fraction of the form a_1/b_1(x) fplus a_2/b_2(x) fplus fdots where a_k are constants and b_k(x) are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for 2∑_k=1^∞ ((−1)^k+1)/(x+2k−1), 2∑_k=0^∞ 1/((x+2k+1)^2), 2∑_k=0^∞ ((−1)^k)/((x+2k+1)^2), ∑_k=1^∞ 1/((x+k)^3), and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of s, including s=5, 7, 9, 11. They do not follow as simple a pattern as Ramanujan's continued fractions.
ER -