Modifications of the Harmonic Series - Department of Mathematics

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

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

for any c > ε > 0. Then x = n =c. So p p x − n < c ε < c= x , so by the isosceles triangle property, 2 p p 2 1 1 x−n εc − = < = ε . 2 n x nx c p p 25
Appendix Code for Mathematica program: This program calculates the sum of the harmonic sub-series defined by removing a specified digit from the denominators of the terms, where the denominators can be written in a specified base. Here, the program is removing the digit 9 from base 10 representations. base = 10; digit = 9; s = Array[0, {31, 5}]; b = Function[z, Sum[x^z, {x, 0, base - 1}] - (digit^z)]; c = Function[{h, z}, ((-1)^z)*((h + z - 1)!)/(z!)/((h - 1)!)]; a = Function[{h, z}, b[z]*c[h, z]/base^(h + z)]; i = 1; j = 1; While[i < 4, n = base^(i - 1); ad = 0; While[n < base^i, If[Length[Cases[IntegerDigits[n,base],digit]]==0, ad=ad+1/n^j] ; n++] s[[i, j]] = ad; ; i++] i = 4; j = 1; ad = Array[0, 5]; k = 1; While[k
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 and 22: In this table, the theorems can mak
Page 23 and 24: We will discuss only a few of the g
Page 25: So n→∞ an+ 1 an p 1 1 a − a =
Page 29 and 30: Bibliography [1] A. J. Kempner, A C

Modifications of the Harmonic Series - Department of Mathematics

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