Problem 49 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. $$ a_{1}=3, r=1.2 $$

Step-by-Step Solution

Answer

The sum does not exist because the series diverges (\( r > 1 \)).

Step 1: Identify the Geometric Series

The given infinite geometric series has a first term \( a_1 = 3 \) and a common ratio \( r = 1.2 \).

Step 2: Determine if the Series Converges

For an infinite geometric series \( a_1 + a_1 r + a_1 r^2 + \cdots \) to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). In this case, \( r = 1.2 \) and \( |1.2| = 1.2 \), which is greater than 1. Thus, the series does not converge.

Step 3: State the Implication

Since the common ratio \( r = 1.2 \) is greater than 1, the infinite geometric series does not have a finite sum. As a result, the sum does not exist.

Key Concepts

Convergence of Series

Understanding whether an infinite geometric series converges or diverges is key to determining its sum. In an infinite geometric series, the series will only converge if the absolute value of the common ratio \(|r|\) is less than 1. This is because the terms in the series must approach zero as the series extends infinitely.
In cases where \(|r| < 1\), the series will have a finite sum. However, if \(|r| \geq 1\), the terms do not diminish to zero, causing the series to diverge and consequently not having a finite sum. For example, with a common ratio of \(1.2\):

• \(|r|\) is greater than 1, meaning the series does not converge.
• Because it diverges, the series does not have a numerical sum.

For students tackling these exercises, recognizing the common ratio's magnitude is a crucial first step to solving problems related to the convergence of series.

Common Ratio

The common ratio \( r \) in a geometric series is the factor by which we multiply each term to get to the next term. It dictates the series' behavior by determining whether it converges or diverges.
A few key points about the common ratio include:

• When \(|r| < 1\), each successive term gets smaller and the series converges.
• When \(|r| = 1\), every term is constant, which typically results in divergence unless the first term is zero.
• When \(|r| > 1\), the terms grow larger, leading to divergence as seen in the example where \(r = 1.2\).

Students should always check the magnitude of \( r \) first in order to ascertain the path of the series and whether there is any potential sum.

Geometric Series Sum

The sum of an infinite geometric series can be calculated using a specific formula, but it applies only if the series converges. For an infinite geometric series with a first term \( a_1 \) and a common ratio \( r \), the formula to find the sum \( S \) is:
\[S = \frac{a_1}{1 - r}\]
This formula is valid under the condition that the series actually converges, meaning \(|r| < 1\).
However, in the case where the common ratio is \( r = 1.2 \), this formula cannot be used. Since 1.2 is greater than 1, the series diverges, making the sum nonexistent.
Understanding when and how to use the geometric series sum formula can save time and effort while solving these problems.