Problem 45 - A Graphical Approach to College Algebra

Question

Write the sum of each geometric series as a rational number. $$0.378+0.000378+0.000000378+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{14}{37} \).

Step 1: Identify the First Term

The first term of the geometric series is the first number in the sequence. Here, the first term (denoted as \( a \)) is \( 0.378 \).

Step 2: Determine the Common Ratio

To find the common ratio (denoted as \( r \)), we divide the second term by the first term. So, the common ratio is \( r = \frac{0.000378}{0.378} = 0.001 \).

Step 3: Check the Common Ratio Condition

For an infinite geometric series to sum to a finite number, the condition \( |r| < 1 \) must be satisfied. Since \( 0.001 < 1 \) holds true, the series can be summed.

Step 4: Apply the Formula for the Sum of an Infinite Geometric Series

The sum of an infinite geometric series can be found using the formula \( S = \frac{a}{1-r} \). Here, \( a = 0.378 \) and \( r = 0.001 \), so substitute these values into the formula: \[ S = \frac{0.378}{1 - 0.001} = \frac{0.378}{0.999} \].

Step 5: Simplify the Fraction

To express \( \frac{0.378}{0.999} \) as a rational number, multiply both the numerator and the denominator by 1000 to eliminate the decimals: \( \frac{378}{999} \). Simplify by dividing both by 3 to get \( \frac{126}{333} \). Simplifying further by another factor of 3, we get \( \frac{42}{111} \). Finally, simplify by the greatest common factor 3 to obtain \( \frac{14}{37} \).

Key Concepts

Common Ratio

In any geometric series, the common ratio is a key element that defines how each term relates to its predecessor. Understanding the common ratio helps to identify patterns and determine the behavior of the series as a whole.

To find the common ratio, you divide any term in the sequence by the term preceding it. For example, if the first term is 0.378 and the second term is 0.000378, you would calculate:

• \( r = \frac{0.000378}{0.378} = 0.001\) .

The common ratio is consistent throughout the series. This means every subsequent term can be obtained by multiplying the current term by 0.001.

A key property of the common ratio is its magnitude. For an infinite geometric series to have a sum, the absolute value of the common ratio must be less than one, ensuring that the terms progressively get smaller.

Infinite Series

An infinite series is a sum of endless terms. Specifically, for geometric series, this involves endlessly adding terms that follow a consistent pattern, dictated by the common ratio.

The intriguing aspect of some infinite series is that they can actually converge to a specific number. This phenomenon occurs when each new term contributes progressively less to the total sum.

For the series in question, because the common ratio \( r = 0.001 \) is less than 1, the series is convergent. This means it will add up to a finite number. However, if \( |r| \) was equal to or greater than 1, the series wouldn't converge and wouldn't have a finite sum.

Sum Formula

The sum formula of an infinite geometric series is a great tool for finding the sum without tediously adding endless terms.

The formula is given by:

• \( S = \frac{a}{1-r} \)

where \( a \) is the first term and \( r \) is the common ratio. Applying this to our series, where \( a = 0.378 \) and \( r = 0.001 \), the sum would be:

• \( S = \frac{0.378}{1 - 0.001} = \frac{0.378}{0.999} \)

To express this sum as a rational number, you simplify \( \frac{0.378}{0.999} \), multiply by 1000 to clear the decimals, and simplify further:

• Start with: \( \frac{378}{999} \)
• Simplify by dividing both numerator and denominator by 3: \( \frac{126}{333} \)
• Simplify again by dividing by 3: \( \frac{42}{111} \)
• Final simplification gives \( \frac{14}{37} \).

This gives the final rational form of the geometric series' sum.