Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$12,24,48,96, \dots$$

Step-by-Step Solution

Answer

The series does not converge as \(r = 2\).

Step 1: Understand the problem

We are given an infinite geometric sequence with the terms 12, 24, 48, 96, and so on. We need to find the common ratio \(r\) and determine if the sum of the sequence converges.

Step 2: Identify the first term \(a\)

In a geometric sequence, the first term is often denoted as \(a\). Here, the first term \(a = 12\). This value will help in calculating the common ratio \(r\).

Step 3: Find the common ratio \(r\)

The common ratio \(r\) can be found by dividing the second term by the first term. Thus, \(r = \frac{24}{12} = 2\).

Step 4: Check for convergence

The sum of an infinite geometric series converges if the absolute value of the common ratio is less than 1, i.e., \(|r| < 1\). Here, \(r = 2\), and \(|2| \geq 1\).

Step 5: Conclusion on convergence

Since \(|r| = 2\), which is greater than 1, the series does not converge.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial component. It is the factor by which we multiply each term to get the next term in the sequence.
To find the common ratio, simply divide a term by the preceding term.

• For example, in the sequence 12, 24, 48, 96, the common ratio \(r\) is \(\frac{24}{12} = 2\)
• The same ratio can be found by dividing 48 by 24 or 96 by 48, as consistent multiplication confirms the ratio.

Thus, here the common ratio is \(r = 2\), indicating exponential growth.
Understanding this common ratio helps us analyze the behavior of the sequence.

Convergence

Convergence in the context of an infinite geometric series refers to whether the series approaches a finite sum.
For a geometric series to converge, its common ratio \( r \) must satisfy:

• \(|r| < 1\)

This condition ensures that as you add infinitely many terms, the series will settle towards a definite value.
In our sequence, the common ratio \(r = 2\) does not meet this criterion since \(|2| \geq 1\).
Therefore, the series does not converge, implying it will not sum to a finite number.

Infinite Series

An infinite series is a summation of an infinite sequence of terms. In mathematics, understanding whether an infinite series converges is essential.
In geometric series specifically:

• The sum becomes meaningful only if it converges.
• The formula for the sum of a convergent infinite geometric series is \( S = \frac{a}{1 - r} \)

Here, "\(a\)" is the first term, and "\(r\)" is the common ratio.
When \( |r| \geq 1 \), like our series with \(r = 2\), the series becomes divergent.
This means it continues to grow indefinitely without approaching a sum, illustrating why convergence is key in infinite series analysis.