12 16) Decimal Expansion [16 Marks, 4 for each part] Note: In this question we are...
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12 16) Decimal Expansion [16 Marks, 4 for each part] Note: In this question we are going to see how to use the MCT to build a decimal expansion for every real number in [0, 1]. a) What is ? k=1 b) Let {b} be a sequence with b, E {0, 1, 2,...,9}. Let In k=1 nondecreasing and bounded above by 1. Note: This shows that {r} converges and that if ß= lim rn, then 3 € [0, 1]. 11-00 c) Let a € [0, 1]. Define a₁ = max{k € {0, 1, 2,...,9}<a}. Then let s₁ = 0≤ 81 ≤a and that a-81 < Sn+1 = 1 10° Next let a2 = max{k € {0, 1, 2,...,9} | 100 ≤a-s1} and let 82 = 10+102. We next proceed recursively in such a manner that if we have defined a₁, a2,..., an and 81, 82,..., Sn, then we define an+1 = = max{k € {0, 1, 2,...,9}|≤a- Sn} and n+1 i) If a = 5, what is a₁? What is s₁? [2 marks] ii) If a = 5, what is as? What is ss? [2 marks] Show that {rn} is 10⁰ d) Given a € [0, 1] and {sn} as defined above, show that lim sn = a. 1140 10 Observe that 12 16) Decimal Expansion [16 Marks, 4 for each part] Note: In this question we are going to see how to use the MCT to build a decimal expansion for every real number in [0, 1]. a) What is ? k=1 b) Let {b} be a sequence with b, E {0, 1, 2,...,9}. Let In k=1 nondecreasing and bounded above by 1. Note: This shows that {r} converges and that if ß= lim rn, then 3 € [0, 1]. 11-00 c) Let a € [0, 1]. Define a₁ = max{k € {0, 1, 2,...,9}<a}. Then let s₁ = 0≤ 81 ≤a and that a-81 < Sn+1 = 1 10° Next let a2 = max{k € {0, 1, 2,...,9} | 100 ≤a-s1} and let 82 = 10+102. We next proceed recursively in such a manner that if we have defined a₁, a2,..., an and 81, 82,..., Sn, then we define an+1 = = max{k € {0, 1, 2,...,9}|≤a- Sn} and n+1 i) If a = 5, what is a₁? What is s₁? [2 marks] ii) If a = 5, what is as? What is ss? [2 marks] Show that {rn} is 10⁰ d) Given a € [0, 1] and {sn} as defined above, show that lim sn = a. 1140 10 Observe that
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16 a 10 aarar ar c i n1 h b Consider V 41 buti 10 9 1 1 1to 5 n Y L 4 9 5 n Thus is 9 ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
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