Question

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{09}=0.090909 \ldots$$

Short answer

Question: Convert the repeating decimal 0.090909... into a fraction. Answer: The repeating decimal 0.090909... can be converted into the fraction 9/99.

Step-by-step solution

Convert the repeating decimal into a geometric series

To represent the repeating decimal as a geometric series, first, let's identify the repeating part, which is 09. We can represent the decimal as a series using the formula: $$0.090909\ldots = 9 \cdot 10^{-2} + 9 \cdot 10^{-4} + 9 \cdot 10^{-6} + \ldots$$ The geometric series is: $$0.090909\ldots = 9(10^{-2} + 10^{-4} + 10^{-6} + \ldots)$$ Notice that this series has the common ratio \(r = 10^{-2}\).

Sum the geometric series

We'll use the formula for the sum of an infinite geometric series: $$S = \frac{a}{1 - r}$$ Where \(S\) is the sum, \(a\) is the first term, and \(r\) is the common ratio. In our case, \(a = 10^{-2}\) and \(r = 10^{-2}\). We plug in the values and get: $$S = \frac{10^{-2}}{1 - 10^{-2}}$$

Multiply the sum by the coefficient

Now, we need to multiply the sum of the geometric series by the coefficient (which is 9 in our case) to get the final sum: $$0.090909\ldots = 9 \cdot S = 9 \cdot \frac{10^{-2}}{1 - 10^{-2}}$$

Simplify the fraction

Next, we have to simplify the fraction: $$0.090909\ldots = \frac{9}{10^2} \cdot \frac{1}{1 - 10^{-2}}$$ Multiply the numerator and denominator by \(10^2\) to clear the denominator: $$0.090909\ldots = \frac{9}{10^2 - 1} = \frac{9}{99}$$

Write the final answer

The repeating decimal \(0.\overline{09}\) can be written as a fraction: $$0.\overline{09} = \frac{9}{99}$$

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For a geometric series to be effectively analyzed, it usually needs a recognizable pattern such as:

• The first term, denoted as \(a\)
• A common ratio, denoted as \(r\)

To express the repeating decimal 0.090909... as a geometric series, we need to look at it as an infinite series where each term is derived by multiplying the previous term by a certain factor (the common ratio). In this case, each subsequent term is the previous term multiplied by \(10^{-2}\). By identifying the repeating pattern — 09 in the tenths and hundredths place — we can express the decimal as:\[0.090909\ldots = 9(10^{-2} + 10^{-4} + 10^{-6} + \ldots)\]This shows clearly how the repeated sequence of the decimal fits into the framework of a geometric series.

Sum of Infinite Series

Calculating the sum of an infinite geometric series like 0.090909... requires understanding a special formula. For an infinite geometric series to have a sum, the absolute value of the common ratio \(r\) must be less than 1.The formula to find the sum \(S\) of such a series is:\[S = \frac{a}{1 - r}\]where \(a\) is the first term and \(r\) is the common ratio. In the given example, the first term \(a = 9 \times 10^{-2}\) and the common ratio \(r = 10^{-2}\). Plugging these into the formula, we calculate:\[S = \frac{10^{-2}}{1 - 10^{-2}}\]This formula helps sum up all the infinitely many terms by accounting for their rapidly decreasing sizes. Hence, multiplying the result by the coefficient 9 accounts for the repeating portion throughout the decimal sequence, ultimately allowing us to express the infinite decimal as a finite fraction.

Converting Decimals to Fractions

Converting a repeating decimal to a fraction involves using the concept and result from the geometric series. Once the infinite series is expressed and summed as described earlier, the repeating decimal can be rewritten as a simple fraction.First, multiply the sum of the infinite series by the coefficient, which represents how many times the fractional unit (formed from the series) repeats. For 0.090909..., this coefficient is 9:\[0.090909\ldots = 9 \cdot \frac{10^{-2}}{1 - 10^{-2}}\]Then, simplify this complex expression by rationalizing it. By multiplying both the numerator and the denominator with the same power of ten that clears the denominators, the fraction simplifies to:\[0.090909\ldots = \frac{9}{99}\]The final step is simplifying \(\frac{9}{99}\). Both the numerator and denominator can be divided by their greatest common divisor, which is 9, yielding:\[\frac{9}{99} = \frac{1}{11}\]Thus, the repeating decimal \(0.\overline{09}\) expressed as a proper fraction is \(\frac{1}{11}\), completing our conversion efficiently.