Modifications of the Harmonic Series - Department of Mathematics

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

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Value Value 25 20 15 10 5 0 4500 4000 3500 3000 2500 2000 1500 1000 Graph 1 Values of Modified Harmonic Series Missing a Given Digit 101201230123401234501234560123456701234567801234567890 500 2 3 4 5 6 7 8 9 10 0 Digit Removed in Base Graph 2 Values of Modified Harmonic Series Missing a Given Digit 1234567890123456789100123456789101101234567891011120 10 11 12 13 Digit Removed in Base 13
3. Convergence and Divergence of Other Modifications Removing terms that contain certain digits is not the only way to modify the harmonic series. In this section we look at one specific example; removing all terms with composite denominators. We will also discuss a method for determining when any given modification converges. 3.1 Reciprocals of Primes We’ve seen that by forbidding a certain digit from the denominators of the harmonic series, the resulting series will converge. At first glance, the prime numbers may seem much rarer, much more spaced out than numbers that do not contain a certain digit. However, this does not mean that we can modify the harmonic series to contain only reciprocals of prime numbers and have it converge. The series 1 ∑ p where p is prime actually diverges. This was first shown by Euler in 1737. Here, we present a proof by contradiction due to Paul Erdös (1938) [12]. PROOF: Assume that the series converges. Then there exists k ∈ such that ∑ i> k i 14 1 1 2 p
Page 1 and 2: THE PENNSYLVANIA STATE UNIVERSITY S
Page 3 and 4: Contents 1. History................
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: Table 2 Base Digit Value Base Digit
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

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