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A Tiling Lemma and Its Application to the Ratio Test for Convergence of Series

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Author(s): James D. Stein Jr. | Linda Ho

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 01;
Issue: 05;
Start page: 300;
Date: 2011;
Original page

Keywords: Tiling | Ratio Test | Banach Fixed-Point Theorem

ABSTRACT
We prove that any collection which tiles the positive integers must contain one of two types of sub-collections. We then use this result to prove a variation of the Ratio Test for convergence of series. This version of the Ratio Test shows the convergence of certain series for which the Root Test (which is known to be more powerful than the conventional Ratio Test) fails. This version of the Ratio Test is also used to prove a version of the Banach Contraction Principle for self-maps of a complete metric space.
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