Problem 19 - Calculus: Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 10 - 2 + 0.4 - 0.08 + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The series is convergent and its sum is \( \frac{25}{3} \).

Step 1: Identify the First Term

The first step in analyzing a geometric series is to identify the first term, which is denoted as \( a \). In this series, the first term is \( 10 \).

Step 2: Determine the Common Ratio

Next, identify the common ratio \( r \) of the series by dividing the second term by the first term: \( r = \frac{-2}{10} = -0.2 \). Confirm this by checking it with the third term: \( r = \frac{0.4}{-2} = -0.2 \). Therefore, the common ratio is \( -0.2 \).

Step 3: Assess Convergence or Divergence

For a geometric series with a common ratio \( r \), the series converges if \( |r| < 1 \). Here, \( |-0.2| = 0.2 < 1 \), so the series is convergent.

Step 4: Calculate the Sum of the Series

The sum \( S \) of an infinitely convergent geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Substitute \( a = 10 \) and \( r = -0.2 \) into the formula: \[ S = \frac{10}{1 - (-0.2)} = \frac{10}{1 + 0.2} = \frac{10}{1.2} = \frac{25}{3} \]. Therefore, the sum of the series is \( \frac{25}{3} \).

Key Concepts

Understanding Convergence in Geometric Series

When we talk about convergence in a geometric series, we are looking to determine if the series adds up to a finite number as the number of terms goes to infinity. Convergence depends on the common ratio, which we'll discuss in the next section.
To check if a series converges, we investigate the absolute value of the common ratio, denoted as |r|. The crucial rule here is that the series will converge if |r| is less than 1.
In the provided series, the common ratio r is \(-0.2\). Since the absolute value \(|-0.2| = 0.2\) is less than 1, the series converges.
This means despite the infinite terms that follow the series, their overall contribution gets smaller and smaller, culminating in a fixed finite sum.

The Importance of the Common Ratio

The common ratio is a fundamental part of a geometric series. It is the factor by which we multiply each term in the series to get the next one.
Understanding how the common ratio influences the series is key:

• If the common ratio \(r = 1\), each term is the same, leading to a straightforward repetition.
• If \(r > 1\), the terms grow larger, causing the series to diverge to infinity.
• If \(r < -1\), the series diverges as terms alternate between large positive and negative values.
• If \(-1 < r < 1\), the series converges, as terms get smaller and approach zero.

In this series, our common ratio is \(-0.2\). This not only shows that the series converges, but it tells us about the behavior of the terms, which flip between positive and negative values while decreasing in magnitude.

Calculating the Sum of a Convergent Series

Once we know that a geometric series is convergent, we can calculate its sum using a specific formula: \[ S = \frac{a}{1 - r} \] Here, \(a\) is the first term and \(r\) is the common ratio.
This formula is powerful as it gives us the complete sum of an infinite series, something made possible only because the series converges.
For the series in the exercise, where the first term \(a = 10\) and the common ratio \(r = -0.2\), applying the formula gives us:

• Compute the denominator: \(1 - (-0.2) = 1 + 0.2 = 1.2\).
• Calculate the sum: \( S = \frac{10}{1.2} = \frac{25}{3} \).

This result, \(\frac{25}{3}\), is the finite sum of the entire series, confirming its convergence and offering a numerical endpoint to the endless sequence of terms.