a. Explanation Given: The number 0.55555555 … which can be viewed as the series 5 ∑ k = 1 ∞ 10 − k . Result used: The geometric series ∑ n = 1 ∞ a r n − 1 = a + a r + a r 2 + ⋯ is convergent if | r | < 1 and its sum is a 1 − r , where a ≠ 0 is the first term and r is the common ratio of the series. Calculation: Consider the geometric series 5 ∑ k = 1 ∞ 10 − k = 5 ∑ k = 0 ∞ 1 10 ( 1 10 ) k with first term of the series as a = 1 10 and common ratio is r = 1 10 < 1 b. To determine To evaluate: The geometric series to obtain a ration value of 0.54545454 … . c. To determine To explain: The method to use geometric series to obtain the form 0. n ` n 2 ⋯ n p ¯ . d. To determine To evaluate: The number 0. 123456789 ¯ by using the method of part (c). e. To determine To prove: The value 0. 9 ¯ = 1

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 8.3, Problem 88E

To determine

To evaluate: The geometric series to obtain a rational value of 0.55555555….

To determine

To evaluate: The geometric series to obtain a ration value of 0.54545454….

To determine

To explain: The method to use geometric series to obtain the form 0.n`n2⋯np¯.

To determine

To evaluate: The number 0.123456789¯ by using the method of part (c).

To determine

To prove: The value 0.9¯=1

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Chapter 8.3, Problem 87E

Chapter 8.3, Problem 89E

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Chapter 8 Solutions

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The geometric series to obtain a rational value of 0.55555555 … .

Solution Summary: The author evaluates the geometric series to obtain a rational value of 0.55555555dots , which can be viewed as the series 5displaystyle undersetn=1

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To evaluate: The geometric series to obtain a rational value of 0.55555555….

To evaluate: The geometric series to obtain a ration value of 0.54545454….

To explain: The method to use geometric series to obtain the form 0.n`n2⋯np¯.

To evaluate: The number 0.123456789¯ by using the method of part (c).

To prove: The value 0.9¯=1

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Chapter 8 Solutions