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Decimal expansionsa. Consider the number 0.555555…, which
Chapter 11, Problem 72AE(choose chapter or problem)
Decimal expansions
a. Consider the number 0.555555..., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k}\). Evaluate the geometric series to obtain a rational value of 0.555555…
b. Consider the number 0.54545454..., which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k}\). Evaluate the geometric series to obtain a rational value of the number.
c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length p, say, \(n_1,\ n_2\ldots n_p,\) where \(n_1,\ldots,\ n_p\) are integers between 0 and 9. Explain how to use geometric series to obtain a rational form of the number.
d. Try the method of part (c) on the number 0.123456789123456789....
e. Prove that \(0 . \overline{9}=1\).
Questions & Answers
QUESTION:
Decimal expansions
a. Consider the number 0.555555..., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k}\). Evaluate the geometric series to obtain a rational value of 0.555555…
b. Consider the number 0.54545454..., which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k}\). Evaluate the geometric series to obtain a rational value of the number.
c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length p, say, \(n_1,\ n_2\ldots n_p,\) where \(n_1,\ldots,\ n_p\) are integers between 0 and 9. Explain how to use geometric series to obtain a rational form of the number.
d. Try the method of part (c) on the number 0.123456789123456789....
e. Prove that \(0 . \overline{9}=1\).
ANSWER:Problem 72AEDecimal expansionsa. Consider the number 0.555555…, which can be viewed as the series . Evaluate the geometric series to obtain a rational value of 0.555555…b. Consider the number 0.54545454…, which can be represented by the series. Evaluate the geometric series to obtain a rational value of the number.Solution:Step 1Considering the number 0.555555… which can be represented as the series .The sum of this geometric series is calculated to get the rational value of the number 0.555555….0.555555…= .Hence the rational value of the number 0.555555… is .