Question

8.3.1 Prove the following: (a) Every infinite series of the form 0O 72 (8.2) using appropriate choices for the coefficients Ibn], with the restrictioin that either bn- 0 or bn 1 is true for every n, represents a number in the interval [0, 1), with the exception that if bn n then the sum of the series is exactly 1. (b) Every real number in the interval [0,1) can be represented using a binary expansion, that is, can be represented by a series of the form (8.2) (c) An expansion of the form (8.2) is said to be terminating if there is some N such that an 0 for all n > N. What property characterizes those numbers in 0,1) which have terminating binary expansions? (d) If a number in [0.1) has a terminating expansion, of the form (8.2), show that it also has exactly one nonterminating expansion. Describe how that non-terminating expansion is constructed (e) Show that the representation of a number in [0,1) by a binary expan sion is unique if and only if the number has no terminating expansion

Answer 1

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