Problem 43 - College Algebra

Question

Find the sum of the infinite geometric series. $$ -1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64} \ldots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \(-\frac{4}{3}\).

Step 1: Identify the first term

The first term of the series is given as \(-1\). This can be directly observed from the series sequence.

Step 2: Identify the common ratio

To find the common ratio \(r\), divide the second term of the series by the first term. Thus, \(r = \frac{-\frac{1}{4}}{-1} = \frac{1}{4}\).

Step 3: Check if the series is convergent

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Here, \(|\frac{1}{4}| = \frac{1}{4}\), which is less than 1, indicating convergence.

Step 4: Apply the sum formula for an infinite geometric series

The sum \(S\) of an infinite geometric series can be calculated using 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 \(a = -1\) and \(r = \frac{1}{4}\) into the formula: \[S = \frac{-1}{1 - \frac{1}{4}} = \frac{-1}{\frac{3}{4}} = -\frac{4}{3}\].

Key Concepts

Common Ratio

In an infinite geometric series, the pattern between consecutive terms is defined by the common ratio, denoted as \(r\). This is a crucial element as it determines the nature and behavior of the series. To find \(r\), take any term in the series and divide it by its preceding term. For the series -1, \(-\frac{1}{4}\), \(-\frac{1}{16}\), \(-\frac{1}{64}\), the common ratio is calculated by dividing the second term by the first: \( r = \frac{-\frac{1}{4}}{-1} = \frac{1}{4} \).

This division shows that each term is one-fourth of its preceding term in absolute terms, as the negative signs cancel each other here. The consistency of this ratio in all consecutive terms is what transforms the group of numbers into a structured geometric series. Recognizing and computing the common ratio accurately is the first major step in analyzing such a series.

Convergence of Series

The notion of convergence is vital in determining whether we can find a finite sum for an infinite geometric series. Convergence of a series refers to the series being limited to a finite value as the number of terms approaches infinity. An infinite geometric series converges if its common ratio's absolute value is less than 1, i.e., \(|r| < 1\).

For the series given, \( r = \frac{1}{4} \), and its absolute value is clearly less than 1. Consequently, this series converges.
When a geometric series converges, it implies that as you keep adding the terms indefinitely, the total sum approaches a specific number. This property is beneficial in problems involving sequences and series, as it assures us that the series has a limit and a calculable sum.

Sum Formula for Geometric Series

To calculate the sum of an infinite geometric series, once we have confirmed convergence, we use a specific formula: \( S = \frac{a}{1-r} \). Here, \(a\) represents the first term of the series, and \(r\) is the common ratio.

For our provided series, \( a = -1 \) and \( r = \frac{1}{4} \). Plugging these into the formula yields: \[ S = \frac{-1}{1 - \frac{1}{4}} = \frac{-1}{\frac{3}{4}} = -\frac{4}{3} \]This operation shows how the infinite series converges to a finite sum of \(-\frac{4}{3}\). By conceptualizing each step—identifying terms, confirming convergence, and methodically applying the formula—we reveal the practicality and clarity in resolving such infinite series.