Modifications of the Harmonic Series - Department of Mathematics

More documents

Recommendations

Info

$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

Abstract While the harmonic series itself diverges, there exist modifications of the series that converge. This paper explores the modifications that are formed by deleting certain terms from the harmonic series. The most common examples are the convergent series formed by removing all of the terms where the denominator contains a certain digit. In addition to a brief history of the subject, we show a method for estimating these ‘forbidden digit’ modifications from above and below. With the aid of a computer program, the convergence values are calculated to several decimal places. We also explore modifications of the harmonic series in different bases and trends among their convergence values. Modifications formed in ways other than forbidding digits are also explored, with the specific example of removing all terms that are reciprocals of composite numbers. Two theorems that serve as convergence tests for modifications of any series are explored and used to test several modifications of the harmonic series in base ten as well as in other bases. In the last section, we move into the p -adic realm, and show that neither the harmonic series, nor any of its sub-series converges in . Finally we examine the ‘harmonic’ sequence and it’s subsequences in , and show that the terms are dense in . p Keywords: Harmonic series, modifications, subseries, convergence 1 p p
Contents 1. History.....................................................................................................................3 1.1 Kempner, Irwin, and Moulton ........................................................................3 1.2 Perlin, Erdos and Niven ..................................................................................4 1.3 Wadhwa and Baillie........................................................................................5 2. Removing Digits .....................................................................................................6 2.1 Estimates from Above and Below ..................................................................6 2.2 “Exact” Values................................................................................................9 2.3 Removing Digits in Other Bases...................................................................10 3. Convergence and Divergence of Other Modifications......................................14 3.1 Reciprocals of Primes ...................................................................................14 3.2 General Conditions for Convergence............................................................15 4. The Harmonic Series in Qp .................................................................................21 4.1 The p-adic Harmonic Series..........................................................................22 4.2 The p-adic Harmonic Sequence....................................................................23 Appendix.....................................................................................................................26 Bibliography ...............................................................................................................28 2
Page 1: THE PENNSYLVANIA STATE UNIVERSITY S
Page 5 and 6: A year later, in 1915, Kempner pose
Page 7 and 8: 2. Removing Digits 2.1 Estimates fr
Page 9 and 10: so on. Then b 2 > 9/10⋅b1. Simila
Page 11 and 12: Table 1 Forbidden Digit Value of Se
Page 13 and 14: Table 2 Base Digit Value Base Digit
Page 15 and 16: 3. Convergence and Divergence of Ot
Page 17 and 18: ∞ ∑ Given a series a , partitio
Page 19 and 20: Table 3 Modification i r i p The di
Page 21 and 22: In this table, the theorems can mak
Page 23 and 24: We will discuss only a few of the g
Page 25 and 26: So n→∞ an+ 1 an p 1 1 a − a =
Page 27 and 28: Appendix Code for Mathematica progr
Page 29 and 30: Bibliography [1] A. J. Kempner, A C

Modifications of the Harmonic Series - Department of Mathematics

Create successful ePaper yourself

Delete template?

Save as template?