Problem 48 - Essential Calculus

Question

(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{9} $$

Step-by-Step Solution

Answer

The sum of the geometric series represented by the repeating decimal \(0.\overline{9}\) is 1.

Step 1: Identify the geometric series

The repeating decimal \(0.\overline{9}\) can be represented as a geometric series where the first term (a) is 0.9, the common ratio (r) is 0.1, and n approaches infinity. Therefore, our geometric series becomes: \(0.9 + 0.09 + 0.009 + \ldots\).

Step 2: Applying the geometric series sum formula

The sum (S) of an infinite geometric series can be calculated using the formula: \( S = \frac{a}{1-r} \), where a is the first term, and r is the common ratio. Substituting our values into this formula, we get: \( S = \frac{0.9}{1 - 0.1} \).

Step 3: Simplify the sum

Simplifying the equation, we find that: S = 1.

Key Concepts

Repeating Decimal

A repeating decimal is a decimal number that has digits that repeat infinitely. Such numbers can look daunting at first, but they actually follow a neat pattern. For example, consider the repeating decimal \(0.\overline{9}\). This means that the number 9 repeats itself forever after the decimal point.

To express a repeating decimal as a series, we need to identify the pattern of repetition. Once we've identified this pattern, we can write it as a sum of terms in a geometric series. This allows us to understand the repeating decimal in a new way and paves the path for further calculations like finding its sum as a fraction.

• The repeating part after the decimal is just a series of numbers. In our case, 9 repeats.
• We can summarize the entire series as \(0.9 + 0.09 + 0.009 + \ldots\).

Sum of Infinite Series

The infinite series that represents our repeating decimal can actually be summed up into a whole number. In mathematics, especially in the context of geometric series, infinite series can be calculated using specific formulas.
When we have a geometric series that repeats infinitely, we use the infinite geometric series sum formula:
\[S = \frac{a}{1-r}\]
Here, \(a\) stands for the first term of the series, and \(r\) is the common ratio. Applying this to our repeating decimal \(0.\overline{9}\):

• First term \(a = 0.9\)
• Common ratio \(r = 0.1\)

We substitute these values into the formula:
\(S = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1\).
Thus, the sum of our series \(0.\overline{9}\) equals \(1\), showing that the repeating decimal \(0.\overline{9}\) actually equals the number \(1\).

Common Ratio

In a geometric series, the common ratio is crucial as it defines how each term progresses to the next. For \(0.\overline{9}\), the common ratio \(r = 0.1\). This means that each subsequent term in the series is 0.1 times the previous one.

To understand why the common ratio is so important:

• It shows the regular pattern of multiplication that allows the series to consistently approach a particular value as the terms progress.
• The value of \(r\) determines how quickly the series converges to its sum; in our case, it converges to 1.

A common ratio between 0 and 1, like 0.1 in our example, ensures that the terms get smaller and smaller, making it possible to sum an infinite series and get a finite number.
The concept of a common ratio helps demystify how something infinite can result in a definitive value.