Problem 46 - A Graphical Approach to College Algebra

Question

Write the sum of each geometric series as a rational number. $$0.297+0.000297+0.000000297+\cdots$$

Step-by-Step Solution

Answer

The sum of the series as a rational number is \( \frac{3}{11} \).

Step 1: Identify the first term and common ratio

The given series is a geometric series where the first term \( a = 0.297 \) and the series can be written as \( 0.297 + 0.000297 + 0.000000297 + \, \cdots \). To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{0.000297}{0.297} \).

Step 2: Calculate the common ratio

Calculate \( r \) by simplifying \( \frac{0.000297}{0.297} = \frac{297 \times 10^{-6}}{297 \times 10^{-3}} = 10^{-3} = 0.001 \). Therefore, the common ratio \( r = 0.001 \).

Step 3: Use the sum formula for an infinite geometric series

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. Since \( \mid r \mid < 1 \), the sum exists.

Step 4: Plug values into the formula

Substitute \( a = 0.297 \) and \( r = 0.001 \) into the sum formula: \( S = \frac{0.297}{1 - 0.001} = \frac{0.297}{0.999} \).

Step 5: Simplify the expression

Simplify \( \frac{0.297}{0.999} \) by multiplying numerator and denominator by 1000 to eliminate the decimals: \( \frac{297}{999} \).

Step 6: Express as a rational number

Simplify \( \frac{297}{999} \) by finding the greatest common divisor, which is 3. Thus, \( \frac{297}{999} = \frac{297 \div 3}{999 \div 3} = \frac{99}{333} = \frac{33}{111} = \frac{3}{11} \) after simplifying further.

Key Concepts

First Term

In a geometric series, the first term is crucial because it represents the starting point of the sequence. The first term, denoted by \( a \), is the initial number from which subsequent terms are calculated by repeatedly multiplying by the common ratio. For the series given in the exercise, the first term is \( a = 0.297 \). Understanding this helps to set the stage for calculating any future terms or for using formulas to find the sum of the series.
The first term is not only important for identifying the sequence, but it also plays a central role in calculating the sum of the series, especially when dealing with an infinite series like in this exercise.

Common Ratio

The common ratio \( r \) in a geometric series is the factor by which we multiply one term to get to the next. It's a key element that defines how quickly the sequence grows or shrinks.
To find the common ratio, divide any term in the series by its preceding term. For example, in our exercise: \( r = \frac{0.000297}{0.297} = 0.001 \).
The common ratio is essential for understanding the behavior of the series—whether it converges or diverges. A geometric series converges if its common ratio is between \(-1\) and \(1\). In this exercise, since \( r = 0.001 \) which lies within this range, the series converges.

Infinite Series

An infinite geometric series is a sequence of numbers in which the terms continue indefinitely. Each term is generated by multiplying the previous term by the common ratio.
In our exercise, the series \(0.297 + 0.000297 + 0.000000297 + \cdots\) continues without end. The beauty of an infinite series in a geometric context is that even though the series has infinitely many terms, it can have a finite sum.
This property is achievable when the absolute value of the common ratio is less than one, making it possible to use special formulas to find the sum despite the infinite nature of the sequence.

Sum Formula

To find the sum of an infinite geometric series, we rely on a particular sum formula: \( S = \frac{a}{1-r} \).
This formula is applicable when the common ratio \( r \) is between \(-1\) and \(1\), which ensures the series converges to a finite value.

• The numerator \( a \) represents the first term of the series.
• The denominator \( 1-r \) represents the leftover from 'taking out' the infinite multiplication component since \( r < 1 \).

Applying this formula to our exercise, we substitute \( a = 0.297 \) and \( r = 0.001 \), which gives us \( S = \frac{0.297}{0.999} \).
Simplifying this expression by multiplying both the numerator and denominator by 1000, we eventually express the sum as the rational number \( \frac{3}{11} \). This formula provides an efficient way to handle the sum of an infinite geometric series.