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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4. The Harmonic Series in Qp In the same way the rational numbers are completed to form the field of real numbers, they can also be completed to form the field of p -adic numbers. The process of completing relies upon the use of a norm. If we use the ‘normal’ absolute value, then when we complete we get . If we use a different norm instead, we get a completely different field. The field of p -adic numbers, p , is formed by completing using the following norm: where x p −ord p ( x) ⎧⎪ p if x≠0 = ⎨ , ⎪⎩ 0 if x = 0 t ⎪⎧ the largest t for which p | x if x∈ ord px = ⎨ . ⎪⎩ ord pa− ord pb, if x = a / b, a, b ∈, b ≠0 There are two things to note about this norm. First, p must be prime. Second, the prime numbers each define a different norm, and the fields created with the different primes are not isomorphic to each other. The 2-adic numbers are quite different from the 7-adic numbers. 21
We will discuss only a few of the general properties of p here, specifically, the properties needed throughout this section. For further reading on p -adic numbers see [13] and [14]. From the definition of the norm, we see that it can only take on discreet −2 −1 0 1 2 values. Namely, the norm of a number must be in the set {... p , p ,0, p , p , p ...} . As a result, the following two inequalities are identical: x the set = { x∈ | x ≤ 1} is call the set of p -adic integers. p p p Furthermore, the natural numbers, , are contained in – they are all p -adic integers. Additionally, the inverses of the p -adic integers are again p -adic integers. From these properties, we know that p { } 1 nn∈ ∈ p . Since we are looking at the harmonic series, we will also need the condition for a series to converge in p . Without going into details, the condition is the 1 n n a ∞ ∑ = n a p following: A series with ∈ converges in p if and only if lim →∞ a = 0 . We will also need this fact: For a sequence an to converge in p , n n p it must have lim a − a = 0 . n→∞ n+ 1 n p 4.1 The p-adic Harmonic Series It is interesting to examine the harmonic series in p and compare it to what we see in . To this end, we prove several theorems. 22
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 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: In this table, the theorems can mak
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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