A Tiling Lemma and Its Application to the Ratio Test for Convergence of Series
Affiliation(s)
1 Department of Mathematics California State University Long Beach, CA. 2 Department of Mathematics El Camino College Torrance, CA.
1 Department of Mathematics California State University Long Beach, CA. 2 Department of Mathematics El Camino College Torrance, CA.
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.
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.
Cite this paper
nullJ. Stein Jr. and L. Ho, "A Tiling Lemma and Its Application to the Ratio Test for Convergence of Series," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 300-304. doi: 10.4236/apm.2011.15055.
nullJ. Stein Jr. and L. Ho, "A Tiling Lemma and Its Application to the Ratio Test for Convergence of Series," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 300-304. doi: 10.4236/apm.2011.15055.
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.
[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.