Problem 38 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. \(1-0.5+0.25-\ldots\)

Step-by-Step Solution

Answer

The sum of the series is \( \frac{2}{3} \).

Step 1: Identify the First Term

In a geometric series, the first term is denoted by \( a \). In the series \(1 - 0.5 + 0.25 - \ldots\), the first term \( a \) is 1.

Step 2: Find the Common Ratio

The common ratio \( r \) is found by dividing the second term by the first term. For this series, \( r = \frac{-0.5}{1} = -0.5 \).

Step 3: Check the Convergence Condition

The sum of an infinite geometric series exists only if the common ratio \( r \) satisfies \(|r| < 1\). Here, \(|-0.5| = 0.5 < 1\), so the series converges.

Step 4: Use the Infinite Series Sum Formula

The sum \( S \) of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term, and \( r \) is the common ratio.

Step 5: Calculate the Sum

Substitute the values for \( a \) and \( r \) into the formula: \( S = \frac{1}{1 - (-0.5)} = \frac{1}{1 + 0.5} = \frac{1}{1.5} = \frac{2}{3} \).

Key Concepts

Infinite Series

An infinite series is essentially the sum of an unlimited number of terms in a sequence. Think of it as a way to add numbers forever! However, not every infinite series leads to a neat, finite sum. In mathematics, some infinite series grow endlessly, while others settle into a specific value. These outcomes depend heavily on the structure or rules of the series.

• An infinite geometric series, the focus of our exercise, repeatedly multiplies a starting value by a fixed number, known as the common ratio.
• Our series, for instance, starts at 1 and keeps getting multiplied by \(-0.5\): \(1, -0.5, 0.25, \ldots\).

Recognizing a pattern here is key to understanding whether or how the series will sum up to something simple.

Convergence Condition

For an infinite geometric series to have a finite sum, it must satisfy a specific condition known as the convergence condition. This condition hinges on the common ratio, which must have a magnitude less than one.

• This means the series keeps multiplying by a smaller and smaller factor as it goes on.
• Mathematically, the series converges if \(|r| < 1\), where \(r\) represents the common ratio.

In our example, the common ratio is -0.5. Since \(|-0.5| = 0.5 < 1\), the series meets the convergence condition, confirming that a specific sum can be found. Otherwise, the terms would grow or oscillate indefinitely, making them impossible to neatly sum together.

Common Ratio

The common ratio is the secret ingredient in a geometric series. It tells you how to get from one term to the next. You find it by dividing any term by the one before it.

• In our series \(1, -0.5, 0.25, \ldots\), dividing -0.5 by 1 gives -0.5, and dividing 0.25 by -0.5 also gives -0.5.
• This consistency confirms that our chosen series is indeed geometric.

Such a common ratio shapes the behavior and value of the series. If it's too large (\