Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13-22, use any test developed so far, including any from Section 9.2, to decide about the convergence or divergence of the series. Give a reason for your conclusion. \(\sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k}\)

Step-by-Step Solution

Answer

The series converges because the common ratio \( \frac{3}{\pi} < 1 \).

Step 1: Identify the Type of Series

The given series is \( \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} \). This is a geometric series with common ratio \( r = \frac{3}{\pi} \).

Step 2: Determine the Common Ratio

In a geometric series \( \sum_{k=1}^{\infty} ar^{k} \), the common ratio \( r \) is crucial for convergence. Here, \( r = \frac{3}{\pi} \).

Step 3: Check the Condition for Convergence of Geometric Series

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). Calculate \( |\frac{3}{\pi}| \).

Step 4: Calculate the Numerical Value

Calculate \( |\frac{3}{\pi}| \approx 0.9549 \). Since \( |\frac{3}{\pi}| < 1 \), the series converges.

Step 5: Conclusion about Convergence

Since \( |\frac{3}{\pi}| < 1 \), the geometric series \( \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} \) converges.

Key Concepts

Convergence of Series

The concept of convergence is fundamental when dealing with infinite series. An infinite series is essentially the sum of infinitely many terms. But instead of adding them up and going on forever, we need to decide whether the series will settle to a specific number as more terms are added or not. This is what we mean by convergence. The series converges if, by adding enough terms, we can get as close as we want to a certain value, called the sum of the series.
On the other hand, if no such value exists, or the series keeps growing endlessly, we say it diverges. Understanding this distinction helps determine if an infinite series has a meaningful sum. In our exercise, we determined the convergence of a geometric series using the common ratio and its properties. This ensures that even when dealing with infinitely many terms, we have a reliable method to find convergence.

Geometric Series Test

The geometric series test is a quick and reliable way to determine whether a given geometric series converges or diverges. A geometric series is a series of the form \[ S = a + ar + ar^2 + ar^3 + \ldots \] where each term after the first is the result of multiplying the previous term by a fixed number, known as the common ratio, denoted by \( r \). The geometric series test states that a geometric series converges if the absolute value of the common ratio, \( |r| \), is less than 1. If \( |r| \geq 1 \), the series diverges. This principle is straightforward but very powerful.

• Check if \( |r| < 1 \) for convergence.
• If \( |r| \geq 1 \), the series will not converge.

Applying this test, the exercise showed the series \( \sum_{k=1}^{\infty}(\frac{3}{\pi})^{k} \) converges since \( \left| \frac{3}{\pi} \right| \approx 0.9549 < 1 \). This means the series sum approaches a finite number as more terms are added.

Common Ratio in Series

In a geometric series, the common ratio \( r \) is a key element that tells us both the nature and behavior of the series. It is the constant factor by which each term of the series is obtained from the previous one. In mathematical terms, it is calculated as the ratio of any term to its preceding term. For geometric progression, once the first term \( a \) is known, every subsequent term can be determined if you know \( r \). For the series \( \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} \), the common ratio is \( \frac{3}{\pi} \). Knowing the common ratio is essential for conducting a convergence test.

• The common ratio determines the rate of growth or decay of the series.
• When \( r \) is less than 1, in absolute terms, terms diminish, leading to potential convergence.

A common ratio with an absolute value less than one ensures that each subsequent term is smaller in magnitude, thus making the series converge. This understanding aids in predicting the behavior of any geometric series you encounter.