Problem 20 - Calculus: Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 2 + 0.5 + 0.125 + 0.03125 + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The series is convergent, and its sum is \( \frac{8}{3} \).

Step 1: Identify the First Term

The first term of the series is the initial term, which in this case is 2.

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term, \( r = \frac{0.5}{2} = 0.25 \). Double-check by dividing the third term by the second term, \( \frac{0.125}{0.5} = 0.25 \). Thus, the common ratio is 0.25.

Step 3: Check for Convergence

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). Here, \( |0.25| = 0.25 \), which is less than 1, so the series is convergent.

Step 4: Calculate the Sum of the Series

The sum of an infinite 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 the values: \( S = \frac{2}{1 - 0.25} = \frac{2}{0.75} = \frac{8}{3} \).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Unlike arithmetic series, which involve addition or subtraction, geometric series involve consistent multiplication between terms.

In the exercise, the series given is:

• 2 (first term)
• 0.5
• 0.125
• 0.03125
• ...

This series is geometric because each term results from multiplying the previous term by 0.25. Recognizing a series as geometric is crucial because it lets us apply specific rules to find out whether the series converges or diverges, and if it converges, it allows us to calculate the sum of all its terms.

Common Ratio

The common ratio, denoted as \( r \), is the factor by which each term of a geometric series is multiplied to obtain the subsequent term. Finding the common ratio helps us understand how the series progresses.

To identify \( r \) in our series, we can divide any term by its preceding term. For instance:

• Calculating between the first two terms gives \( r = \frac{0.5}{2} = 0.25 \)
• Re-checking between the second and third terms also results in \( \frac{0.125}{0.5} = 0.25 \)

Knowing the common ratio is essential because it determines whether a geometric series converges or diverges. Specifically, a series will converge if the absolute value of the common ratio is less than 1, \( |r| < 1 \). In this case, our \( r = 0.25 \), which satisfies this condition, indicating convergence.

Infinite Series

An infinite series is a series that continues indefinitely, with an infinite number of terms. In mathematics, understanding the behavior of infinite series is vital in applications across different fields.

In our exercise, we explore an infinite geometric series, where the terms go on without end. For such series, identifying convergence is crucial. An infinite geometric series will only have a finite sum, making it convergent, if its common ratio's absolute value is smaller than one, \( |r| < 1 \). If this criterion is met, as in our case, it means despite going on forever, the terms of the series keep getting smaller enough, so the total sum remains finite and calculable.

Sum of a Series

The sum of a series, particularly an infinite geometric series, is a key concept. When a series converges, it means we can calculate the total sum of all its infinite terms using a special formula. This sum is finite only when the common ratio offers conditions for convergence.

For a convergent geometric series, the sum \( S \) can be calculated using the formula:\[ S = \frac{a}{1 - r} \]where \( a \) is the first term, and \( r \) is the common ratio. Applying this formula to our series:

• First term \( a = 2 \)
• Common ratio \( r = 0.25 \)
• Sum \( S = \frac{2}{1 - 0.25} = \frac{2}{0.75} = \frac{8}{3} \)

This calculation shows that the infinite series has a total sum of \( \frac{8}{3} \), highlighting the power of the geometric series formula in assessing and finding the sums of infinite sequences.