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Chapter 9: Problem 43

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{1}=0.111 \ldots$$

Short Answer

Expert verified
Answer: The repeating decimal 0.111... can be represented as the fraction 1/9.

Step by step solution

01

1. Write the repeating decimal as a geometric series

To represent the given repeating decimal \(0.\overline{1}\) as a geometric series, we can express it as an infinite sum of terms with a common ratio of 0.1. This is because each term in the decimal representation is 1 moved one place to the right, which is the same as multiplying by 0.1. Therefore, we can write the repeating decimal as follows: $$0.\overline{1} = 0.1 + 0.01 + 0.001 + \ldots$$ Now, we have the geometric series with the first term \(a_1 = 0.1\) and a common ratio \(r = 0.1\).
02

2. Use the sum formula for an infinite geometric series to find the sum

For an infinite geometric series with a common ratio \(|r| < 1\), the sum can be found using the formula: $$S = \frac{a_1}{1 - r}$$ In our case, the first term \(a_1 = 0.1\) and the common ratio \(r = 0.1\). Plugging these values into the formula, we get: $$S = \frac{0.1}{1 - 0.1} = \frac{0.1}{0.9}$$
03

3. Convert the decimal into a fraction

Now, we have the sum of the geometric series as a decimal: $$S = \frac{0.1}{0.9}$$ To convert the decimal into a fraction (a ratio of two integers), we can multiply the numerator and the denominator by 10 to eliminate the decimals: $$S = \frac{0.1 \times 10}{0.9 \times 10} = \frac{1}{9}$$ So, the repeating decimal \(0.\overline{1}\) can be represented as the fraction \(\frac{1}{9}\).

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Most popular questions from this chapter

Determine whether the following series converge or diverge. $$\sum_{k=1}^{\infty} \frac{2^{k}+3^{k}}{4^{k}}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty}(-1)^{k} e^{-k}$$

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=2 a_{n}\left(1-a_{n}\right) ; a_{0}=0.3, n=0,1,2, \dots$$

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1}\), for \(n=1,2,3, \ldots,\) where \(f_{0}=0, f_{1}=1 .\) Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}},\) the ratio of the successive terms of the sequence. Provide evidence that \(\varphi=(1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)$$

Determine whether the following series converge or diverge. $$\sum_{k=0}^{\infty} \frac{10}{k^{2}+9}$$
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