Problem 8 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the First Term (a) and Common Ratio (r)

The first term of the given series is denoted by \( a \). Here, \( a = 2 \). To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{0.5}{2} = 0.25 \). Check the ratio between subsequent terms to confirm that the series is indeed geometric: \( \frac{0.125}{0.5} = 0.25 \) and \( \frac{0.03125}{0.125} = 0.25 \). Thus, the common ratio \( r = 0.25 \).

Step 2: Check for Convergence

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

Step 3: Calculate the Sum of the Convergent Geometric Series

The sum \( S \) of an infinite convergent geometric series is given by the formula: \( S = \frac{a}{1 - r} \). Substitute the values of \( a \) and \( r \) into the formula: \( S = \frac{2}{1 - 0.25} = \frac{2}{0.75} = \frac{8}{3} \). Thus, the sum of the series is \( \frac{8}{3} \).

Key Concepts

Series Convergence

The concept of series convergence is fundamental in understanding how infinite series behave. In mathematics, particularly in calculus, a series is said to converge if the sequence of its partial sums approaches a finite limit as more terms are added. Convergence gives us valuable information about whether the series will sum up to a specific number or continue increasing without bound.

With geometric series, convergence is determined by the value of the common ratio, denoted as \( r \). This value plays a critical role in the series' behavior.

• If \(|r| < 1\), the geometric series converges, meaning it sums up to a finite value.
• If \(|r| \geq 1\), the series diverges, indicating it does not sum up to a finite amount but instead grows indefinitely.

For example, in the series \(2 + 0.5 + 0.125 + 0.03125 + \cdots\), the common ratio \(|r| = 0.25\), which is less than 1. Therefore, this series converges. Knowing whether a series converges helps mathematicians and scientists to work with infinite series in practical applications without fear of unending calculations.

Sum of Infinite Series

When a geometric series is convergent, it means you can actually calculate its sum, even though there are infinitely many terms. This is one of the fascinating aspects of mathematics, allowing us to consider the collective value of an infinite number of elements.

The formula used to find the sum \( S \) of a converging geometric series is:

• \( S = \frac{a}{1 - r} \)

where \( a \) is the first term, and \( r \) is the common ratio of the series. This formula gives a direct way to find the sum without manually summing each term. It is only applicable when the series is proven to converge, i.e., \(|r| < 1\).

Using our example series with \( a = 2 \) and \( r = 0.25 \):

• Sum: \( S = \frac{2}{1 - 0.25} = \frac{2}{0.75} = \frac{8}{3} \)

Thus, even though there are infinitely many terms, their combined value is finite and equal to \( \frac{8}{3} \). This illustrates how powerful the concept of series convergence is in simplifying complex problems.

Common Ratio

The common ratio is a fundamental characteristic of geometric series that defines its successive terms. Understanding the common ratio helps to identify whether a sequence of numbers forms a geometric series.

For a series to be geometric, each term should be the previous term multiplied by a constant, known as the common ratio. It's calculated by dividing a term by its preceding term:

• \( r = \frac{a_{n+1}}{a_n} \)

where \( a_{n+1} \) represents any term in the series and \( a_n \) the previous one. This ratio should be consistent across the series.

In our problem, the common ratio \( r \) was calculated to be \( 0.25 \), confirming the series as geometric. Determining this ratio enabled us to check for series convergence, ensuring the sum calculation was valid. Recognizing the common ratio is thus crucial not only for identifying the nature of a series but also for calculating sums when applicable.