Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Write the sum of each geometric series as a rational number. (See Exercise 44.) $$0.7+0.07+0.007+0.0007+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{7}{9} \).

Step 1: Identify the Series

The given series is a geometric series with the first term, \( a = 0.7 \), and the subsequent terms \( 0.07, 0.007, 0.0007, \ldots \). The series continues indefinitely.

Step 2: Find the Common Ratio

To find the common ratio \( r \) of the series, divide the second term by the first term: \( r = \frac{0.07}{0.7} \). Simplifying this, \( r = 0.1 \).

Step 3: Use the Sum Formula for Infinite Series

The formula for the sum of an infinite geometric series where \( |r| < 1 \) is \( S = \frac{a}{1-r} \). In this case, \( a = 0.7 \) and \( r = 0.1 \).

Step 4: Plug Values into the Formula

Substitute the known values into the sum formula: \( S = \frac{0.7}{1-0.1} \).

Step 5: Simplify the Expression

Simplify the denominator: \( 1 - 0.1 = 0.9 \). Thus, the sum \( S = \frac{0.7}{0.9} \).

Step 6: Convert to a Rational Number

Simplify \( \frac{0.7}{0.9} \) by multiplying the numerator and the denominator by 10 to eliminate the decimals: \( \frac{7}{9} \). \( 7 \) and \( 9 \) have no common factors, so it is already in its simplest form.

Key Concepts

Infinite Series

An infinite series is a sequence of numbers that continues indefinitely. In mathematics, it often involves adding up all the terms in a sequence, which can go on forever. Understanding infinite series can at first seem daunting because it's hard to imagine adding up infinitely many things. However, with the right mathematical tools, it becomes manageable.

• Geometric series is a type of infinite series.
• It involves multiplying each term by a fixed number to get the next.
• Although the series goes on forever, it can still have a finite sum if certain conditions are met.

For an infinite geometric series to have a finite sum, the absolute value of the common ratio must be less than one. This means that each term is getting smaller and smaller, moving towards zero, allowing the series to "converge" to a specific sum.

Common Ratio

In a geometric series, the common ratio is the factor by which we multiply one term to get to the next term. To find it, you simply divide any term by the one before it. It's essential to identify this correctly as it plays a crucial role in determining the sum of the series. The common ratio can tell us a lot:

• It helps us understand how the series progresses.
• If the common ratio is less than one in absolute value, the series will converge to a finite sum.
• If it is greater than or equal to one, the series will diverge, meaning it goes off to infinity.

In our exercise, the common ratio has been found to be 0.1, indicating a converging series, which means the infinite series can be summed up into a finite number.

Sum of Series

The sum 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. This formula gives us a way to find the total of an otherwise endless sequence of numbers, turning the infinite into the finite.

• The first term \( a \) and the common ratio \( r \) must be known.
• The series must have a common ratio with an absolute value less than one to ensure a finite sum.
• By plugging in the values from the example, we use \( a = 0.7 \) and \( r = 0.1 \), giving us \( S = \frac{0.7}{0.9} \).

This tells us that the infinite series adds up to \( \frac{7}{9} \) when expressed as a rational number.

Fraction Simplification

Fraction simplification is often the final step in expressing a geometric series sum in its simplest form. This process makes the expression easier to understand and work with by reducing it to the smallest possible whole numbers.To simplify a fraction:

• Divide both the numerator and the denominator by their greatest common divisor (GCD).
• If they don't share any common factors other than 1, the fraction is already simplified.
• For example, \( \frac{0.7}{0.9} \) is simplified by multiplying numerator and denominator by 10 to remove the decimals, resulting in \( \frac{7}{9} \).

In our example, the fraction \( \frac{7}{9} \) is already in its simplest form as 7 and 9 share no common factors. Simplifying fractions is crucial as it ensures the final result is neat, precise, and easily interpretable.