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

James D. Stein Jr.^1*, Linda Ho²

¹Department of Mathematics California State University Long Beach, CA.
²Department of Mathematics El Camino College Torrance, CA.

DOI: 10.4236/apm.2011.15055                

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.

Keywords

Tiling, Ratio Test, Banach Fixed-Point Theorem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. R. Jachymski, Schroder, Bernd; Stein, D., James Jr., “A connection between fixed-point theorems and tiling problems,” Journal of Combinatorial Theory, Series A, Vol. 87, No. 2, 1999, pp. 273-286. doi:10.1006/jcta.1998.2960
[2]	W. Rudin, “Principles of Mathematical Analysis,” McGraw-Hill, New York, 1964.