Problem 52 - College Algebra with Modeling and Visualization

Question

Write each national number in the form of an infinite geometric series. $$ \frac{1}{9} $$

Step-by-Step Solution

Answer

\( \frac{1}{9} = \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \dots \)

Step 1. Understand the Geometric Series Formula

Recall the formula for an infinite geometric series: \[ S = \frac{a}{1 - r} \]where \( a \) is the first term and \( r \) is the common ratio. The condition for convergence is that \(|r| < 1\).

Step 2. Set Up the Series

Express \( \frac{1}{9} \) in a similar form \( S = \frac{a}{1 - r} \) to identify \( a \) and \( r \). The goal is to express \( \frac{1}{9} \) based on the sum formula for geometric series.

Step 3. Find a Suitable Expression

Notice that \( \frac{1}{9} = 0.111... \), which is a repeating decimal. This can be expressed symbolically as:\[0.111... = \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \dots\]It represents a geometric series with first term \( a = \frac{1}{10} \) and common ratio \( r = \frac{1}{10} \).

Step 4. Verify the Series Expression

Use the sum formula to verify:\[S = \frac{a}{1 - r} = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9}\]This confirms our series expression is correct.

Key Concepts

geometric series formula

An infinite geometric series is a sum of terms that have a constant ratio between successive terms. It is represented by the formula:\[ S = \frac{a}{1 - r} \]where:

• \( S \) is the sum of the series (if it converges),
• \( a \) is the first term of the series,
• \( r \) is the common ratio between terms.

This formula helps in finding the sum of an infinite series, provided it meets certain conditions. Understanding this formula is crucial in working with geometric series as it provides a systematic method to calculate the sum based on initial term and ratio. Breaking down this formula and recognizing its components will allow you to tackle various problems involving geometric series.

convergence condition

For an infinite geometric series to have a sum, it must meet the convergence condition. This condition states that the absolute value of the common ratio \( r \) must be less than one: \(|r| < 1\).
When this condition is not met, the series does not converge, meaning it does not add up to a finite sum.
Here's why:- If \(|r| < 1\), with each successive term, the size decreases, making it possible to "approach" a certain number; hence, the series converges.- If \(|r| \geq 1\), the terms either stay the same size or grow larger, indicating divergence. This causes the series to either oscillate indefinitely or grow without bound.By ensuring \(|r| < 1\), we can use the geometric series formula effectively to find the sum of an infinite series.

repeating decimal

Repeating decimals are numbers with a set of digits that endlessly repeat in a decimal form. For example, \(0.111\ldots\) is a repeating decimal. Repeating decimals can be expressed as infinite geometric series.
For instance:- \(0.111\ldots\) can be rewritten as the series \(\frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \ldots\).- Here, \(a = \frac{1}{10}\) and \(r = \frac{1}{10}\).This process works because the repeated portions form the same geometric pattern as terms in a series with a constant ratio. The ability to convert repeating decimals into geometric series allows calculations of their exact fractional value through the use of the geometric series sum formula.

sum formula for geometric series

The sum formula for a geometric series helps simplify complex repeating systems into manageable calculations. Given the sum formula:\[ S = \frac{a}{1 - r} \]You start by identifying the first term \( a \) and the common ratio \( r \). For example, with \(0.111\ldots\), we identified the following:

• First term \( a = \frac{1}{10} \)
• Common ratio \( r = \frac{1}{10} \)

Using the formula, the sum of the infinite geometrical series is verified:\[ S = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9} \]This calculation shows that repeating decimals like \(0.111\ldots\) are equivalent to fractions such as \(\frac{1}{9}\). By applying the sum formula, you can transform an infinite series into a finite decimal or fractional representation.