Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$-48,-24,-12,-6, \dots$$

Step-by-Step Solution

Answer

The common ratio \(r\) is \(\frac{1}{2}\). The series converges.

Step 1: Identify the first term

The first term of the infinite geometric sequence, denoted by \(a_1\), is \(-48\).

Step 2: Determine the common ratio

To find the common ratio \(r\) of the geometric sequence, divide the second term by the first term: \( r = \frac{-24}{-48} = \frac{1}{2} \).

Step 3: Check for convergence

For the sum of an infinite geometric sequence to converge, \(|r| < 1\) must be true. Here, \(|r| = \left| \frac{1}{2} \right| = \frac{1}{2} < 1\), so the series converges.

Key Concepts

Common Ratio

In an infinite geometric sequence, the common ratio is a key element. It tells us how each term is derived from the previous one. In the given sequence, -48, -24, -12, -6, ..., the common ratio is determined by dividing any term by the previous term. For example, the second term (-24) divided by the first term (-48) results in a common ratio of \( r = \frac{-24}{-48} = \frac{1}{2} \).

This constant ratio \( \frac{1}{2} \) means that each subsequent term is half the preceding term. This constant multiplicative factor is what defines the pattern of a geometric sequence. Understanding this ratio helps to predict future terms and analyze the overall behavior of the sequence.

Convergence

Convergence in a geometric series refers to whether the sum of all the terms will approach a finite number as the sequence progresses to infinity. For a geometric series to converge, the common ratio \( r \) must satisfy a specific condition: the absolute value of \( r \) should be less than one \(|r| < 1\).

In the sequence -48, -24, -12, -6, ..., the common ratio was calculated as \( \frac{1}{2} \). Since \( \left| \frac{1}{2} \right| < 1 \), this sequence's sum does indeed converge. Convergence is an important concept because it indicates stability in the progression, meaning the series does not grow indefinitely but settles into a limit instead.

Geometric Series

A geometric series is simply the sum of the terms in a geometric sequence. When dealing with infinite geometric series, we often seek whether they converge to a certain sum.

For a geometric series with first term \( a_1 \) and common ratio \( r \), its sum can be calculated using a special formula if \(|r| < 1\):\[S = \frac{a_1}{1 - r}\]In the sequence -48, -24, -12, -6, ..., the first term \( a_1 \) is -48, and \( r \) is \( \frac{1}{2} \). Therefore, this series converges to:\[S = \frac{-48}{1 - \frac{1}{2}} = \frac{-48}{\frac{1}{2}} = -96.\]This formula empowers us to find the limit that the infinite series approaches, facilitating deeper understanding of sequences' behaviors.