Problem 111 - Algebra and Trigonometry

Question

Make up two infinite geometric series, one that has a sum and one that does not. Give them to a friend and ask for the sum of each series.

Step-by-Step Solution

Answer

Sum of series (2, 1, 0.5, ...) is 4. Series (1, 1.5, 2.25, ...) has no sum.

Step 1: Identify the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series where the first term is denoted as \(a\) and the common ratio as \(r\) is given by: \[ S = \frac{a}{1 - r} \] This formula is applicable only if \(|r| < 1\).

Step 2: Create a geometric series that has a sum

Choose values for \(a\) and \(r\) such that the series converges (\(|r| < 1\)). Example: Let \(a = 2\) and \(r = 0.5\). The series is: 2, 1, 0.5, 0.25, 0.125, ... Using the formula: \[ S = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4 \]

Step 3: Create a geometric series that does not have a sum

Choose values for \(a\) and \(r\) such that the series diverges (\(|r| \geq 1\)). Example: Let \(a = 1\) and \(r = 1.5\). The series is: 1, 1.5, 2.25, 3.375, 5.0625, ... This series does not have a sum because \(|r| \geq 1\).

Step 4: Summarize the findings for each series

For the convergent series (2, 1, 0.5, ...), the sum is 4. For the divergent series (1, 1.5, 2.25, ...), there is no sum as the series diverges.

Key Concepts

Convergent Series

A geometric series is called a 'convergent series' if the absolute value of the common ratio is less than 1. This means \(|r| < 1\). When a series is convergent, the terms of the series get smaller and smaller, approaching zero as you proceed through the series. Eventually, the series has a finite sum that can be calculated using a specific formula, even though there are infinitely many terms in the series.

For example, consider the series with first term \(a = 2\) and common ratio \(r = 0.5\). The series is 2, 1, 0.5, 0.25, 0.125, ... . As you can see, each term gets smaller, converging toward zero. The sum of this series can be calculated by the formula:

\[ S = \frac{a}{1 - r}\]

Which in this case is:

\[ S = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4\]

This shows that the series converges to a sum of 4.

Divergent Series

A geometric series is called a 'divergent series' if the absolute value of the common ratio is greater than or equal to 1. This signifies \(|r| \geq 1\). When a series is divergent, the terms of the series do not get smaller over time; instead, they either stay the same size or grow larger. Therefore, the series does not sum to a finite value.

Consider the series with first term \(a = 1\) and common ratio \(r = 1.5\). The series is 1, 1.5, 2.25, 3.375, 5.0625, ... . As you can see, each term keeps increasing. Since \(r\) is greater than 1, the series diverges and does not have a sum.

When dealing with a divergent series, the sum formula cannot be applied. In this example, there is no finite sum that captures the total of all terms in the series.

Sum Formula

The sum formula for an infinite geometric series is a powerful tool in determining the sum of an infinite number of terms in a series. This formula can be used only if the series converges, meaning the absolute value of the common ratio, \(|r|\), is less than 1.

The sum formula is given by:

\[ S = \frac{a}{1 - r}\]

Where \(a\) is the first term of the series and \(r\) is the common ratio.

• If \(|r| <1\), the series converges and has a finite sum.
• If \(|r| \geq 1\), the series diverges and does not have a sum.

For illustration, let's use the series where \(a = 2\) and \(r = 0.5\). By applying the sum formula, we get:

\[ S = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4\]

This clearly shows the sum of an infinite number of terms in the convergent series.