Modifications of the Harmonic Series - Department of Mathematics

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Next, compare each with . The first nine terms of , are each less than a ai 1 a 2 or equal to 1/ 10, so together they are less than 9/ 10. The next nine terms are each less than or equal to 1/ 20 and together less than 9 / 20 = (9 /10) ⋅ (1/ 2) . Continue this process, and it is clear that a2 9/10 1 a < ⋅ . Similarly, 2 a3 81/100 a1 (9 /10) 1 a < ⋅ = ⋅ . 1 Furthermore, (9 /10) . Then we can estimate the modified series from above i− a < ⋅a with i 1 2 3 ⎛ ⎞ 9 ⎛ 9 ⎞ ⎛ 9 ⎞ S < a1+ ⎜ + ⎜ ⎟ + ⎜ ⎟ + ... ⎟a1 = 10a1 ≈27.18 . ⎜10 ⎝10 ⎠ ⎝10 ⎠ ⎟ ⎝ ⎠ Estimating from Below: To estimate the series from below, we use a method similar to the one for estimating from above. Here we partition the series differently and estimate each finite sub-series from the other side. We shift the partitions that we used above by one term, so that 1 1 1 1 b1 = + +⋅⋅⋅+ + 2 3 8 10 1 1 1 1 b2 = + +⋅⋅⋅+ + 11 12 88 100 . 1 1 1 1 b3 = + +⋅⋅⋅+ + 101 102 888 1000 We divide the terms of b again into groups of nine, and compare them to the 2 terms in b . The first group consists of 1/11+ 1/12 + ... + 1/18 + 1/ 20 . Each of these 1 terms is greater than or equal to 1/ 20, and they are together greater than 9/ 20. Likewise, the next nine terms are greater than or equal to 7 9 / 30 = (9 /10) ⋅ (1/ 3) , and
so on. Then b 2 > 9/10⋅b1. Similarly, 1. We can see, then, b ⋅ 2 b3 > 81/100 ⋅ b1 = (9 /10) 1 that (9 /10) ⋅b . So we can estimate the series from below by i− b > i 1 2 3 ⎛ ⎞ 9 ⎛ 9 ⎞ ⎛ 9 ⎞ S > 1 + b1+ ⎜ + ⎜ ⎟ + ⎜ ⎟ + ... ⎟= 1+ 10b1 ≈19.17 . ⎜10 ⎝10 ⎠ ⎝10 ⎠ ⎟ ⎝ ⎠ We have found that 19.17 < S < 27.18. We can make the bounds closer by comparing the parts of the series to and instead of and . In this case, the b estimation from above becomes a2 2 b 1 a 1 2 3 ⎛ ⎞ 9 ⎛ 9 ⎞ ⎛ 9 ⎞ S < a1+ a2 + + + + ... a2 = a1+ 10a2 ≈23.3 ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ 10 ⎝10 ⎠ ⎝10 ⎠ ⎟ ⎝ ⎠ The estimation from below follows similarly, and we get the following estimates for the modified series: 22.4 < S < 23.3. Notice that these estimates are much closer together than the previous ones. If we keep going, and compare everything to and instead, then we have the following b bounds: a3 3 22.8 < S < 23.0. The bounds are very close together now. Going further, comparing the terms to and or and will not yield much improvement. b b4 5 a 5 This method can be used without modification for removing, or forbidding, other digits from the series. With some small changes, it can also be used for finding bounds on a series modified by forbidding digits from the denominators in different 8 a4
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: 2. Removing Digits 2.1 Estimates fr
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

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