Problem 59 - College Algebra with Modeling and Visualization

Question

Write the sum of each geometric series as a rational number. $$0.45+0.0045+0.000045+\cdots$$

Step-by-Step Solution

Answer

The sum is \( \frac{5}{11} \).

Step 1: Identify the first term (a)

In a geometric series, the first term is the initial number in the sequence. Here, the first term \( a \) is the first number, which is \( 0.45 \).

Step 2: Determine the common ratio (r)

The common ratio \( r \) is the factor by which each term is multiplied to yield the next term. Divide the second term by the first: \( r = \frac{0.0045}{0.45} = 0.01 \).

Step 3: Recognize the series is infinite

Since the series continues indefinitely as indicated by the '...', it is an infinite geometric series.

Step 4: Use the formula for the sum of an infinite geometric series

The formula for the sum \( S \) of an infinite geometric series is given by \( S = \frac{a}{1-r} \), where \( 0 < r < 1 \).

Step 5: Substitute values into the formula

Substitute \( a = 0.45 \) and \( r = 0.01 \) into the infinite geometric series sum formula: \( S = \frac{0.45}{1 - 0.01} \).

Step 6: Calculate the sum

Simplify the expression: \( S = \frac{0.45}{0.99} \). Divide the numerator by the denominator to obtain \( S = \frac{45}{99} \).

Step 7: Simplify the fraction

Reduce \( \frac{45}{99} \) to its simplest form by dividing the numerator and denominator by their greatest common divisor (9): \( \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \).

Key Concepts

Understanding Infinite Series

An infinite series is a sum of infinitely many numbers. Imagine starting with one number and then adding another, and another, forever. This might sound mathematically daunting. However, in many cases, we can calculate the sum of an infinite series using specific formulas. In geometric series like the exercise above, the series will converge to a particular value if the common ratio (more on this later) is between -1 and 1.
Convergence and Divergence

• Convergence: When an infinite series approaches a specific finite value.
• Divergence: When an infinite series does not approach any value, making it impossible to find a sum.

Here, the series converges since its common ratio is less than one, allowing us to calculate the sum precisely.

Rational Numbers Explained

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. They have the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). In simple terms, these are numbers that can be written as fractions. For instance, 0.5 is a rational number because it equals \( \frac{1}{2} \).
In our problem, when solving the geometric series, we arrive at \( \frac{5}{11} \), which is a rational number because it is a fraction involving two integers.
Properties of Rational Numbers

• They can be positive, negative, or zero.
• They include whole numbers, because any whole number \( n \) can be written as \( \frac{n}{1} \).
• The decimal representation may terminate or repeat. For example, 0.333... is fractional as \( \frac{1}{3} \).

The Role of Common Ratio in Geometric Series

The common ratio in a geometric series is the consistent factor by which each term is multiplied to obtain the next term. Calculating this is essential in determining the behavior and sum of the series.
In our exercise, the common ratio is found by dividing the second term by the first, which results in a common ratio of 0.01.
Importance of Common Ratio

• If \( |r| < 1 \), the series converges and we can find its sum using the formula \( S = \frac{a}{1-r} \).
• If \( |r| \geq 1 \), the series diverges, making it approach infinity or oscillate.

Understanding the common ratio helps determine if and how a geometric series can be summed, highlighting its vital role in mathematical problem-solving.