Problem 24 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{1}{3}+\frac{2}{9}+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\cdots\)

Step-by-Step Solution

Answer

The series is convergent with a sum of \(\frac{3}{4}\).

Step 1: Identify the Format of the Series

Observe that the series given is an infinite series. It consists of a repeating pattern with two types of terms: \(\frac{1}{3^n}\) and \(\frac{2}{3^n} \) for \( n = 1, 2, 3,\ldots \). Therefore, the series can be expressed as the sum of two separate geometric series: \( S_1 = \sum_{n=1}^{\infty} \frac{1}{3^n} \) and \( S_2 = \sum_{n=1}^{\infty} \frac{2}{3^{2n}} \).

Step 2: Analyze the First Sub-Series \( S_1 \)

The series \( S_1 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \) is a geometric series with first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \). A geometric series converges if \( |r| < 1 \). Since \( |\frac{1}{3}| < 1 \), \( S_1 \) converges. The sum of an infinite geometric series is given by \( \frac{a}{1-r} \). Thus, \( S_1 = \frac{\frac{1}{3}}{1-\frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \).

Step 3: Analyze the Second Sub-Series \( S_2 \)

The series \( S_2 = \frac{2}{9} + \frac{2}{81} + \frac{2}{729} + \cdots \) is also a geometric series with first term \( a = \frac{2}{9} \) and common ratio \( r = \frac{1}{9} \). Since \( |\frac{1}{9}| < 1 \), \( S_2 \) converges. The sum of this series is \( \frac{a}{1-r} \), so: \( S_2 = \frac{\frac{2}{9}}{1-\frac{1}{9}} = \frac{\frac{2}{9}}{\frac{8}{9}} = \frac{2}{8}=\frac{1}{4} \).

Step 4: Calculate the Total Sum of the Series

The total sum of the original series is the sum of \( S_1 \) and \( S_2 \): \( \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \). Therefore, the series is convergent and its sum is \( \frac{3}{4} \).

Key Concepts

Geometric Series

A geometric series is one of the fundamental types of series in mathematics, where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." Geometric series have the form: \[ a + ar + ar^2 + ar^3 + \cdots \] Here, "\( a \)" represents the first term and "\( r \)" is the common ratio. Understanding the basics of geometric series allows us to determine if such a series converges or diverges and to calculate its sum if it converges.
Characteristics of a geometric series include:

• The presence of a constant ratio \( r \) between consecutive terms.
• The series can be either finite or infinite.

A classic example is the first sub-series from the given exercise: \( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \), where the common ratio \( r \) is \( \frac{1}{3} \).
Recognizing geometric series helps to effectively apply techniques for finding sums and testing convergence.

Convergence Criteria

To determine whether a geometric series converges, we rely on a simple yet powerful rule: a geometric series converges if and only if the absolute value of the common ratio \( |r| < 1 \). If this condition is met, the series converges to a finite sum. The formula for the sum of an infinite convergent geometric series is: \[ S = \frac{a}{1-r} \] Here, "\( a \)" is the first term and "\( r \)" is the common ratio. This formula only holds when \( |r| < 1 \).
Consider our exercise example, in both sub-series \( S_1 \) and \( S_2 \):

• For \( S_1 \) with \( r = \frac{1}{3} \), \( |r| < 1 \), hence it converges.
• For \( S_2 \) with \( r = \frac{1}{9} \), again \( |r| < 1 \), so it also converges.

This criterion is incredibly useful for identifying convergent geometric series and is essential to solving problems that involve summing such series.

Infinite Series

An infinite series is a sum of infinitely many terms. It has the general form: \[ a_1 + a_2 + a_3 + \cdots \] Unlike finite series, infinite series continue indefinitely. The main challenge is determining if the series sums to a finite number (convergent) or grows without bound (divergent).
Infinite series are at the heart of many mathematical analyses and have significant applications across different fields, including physics and engineering. In the original exercise, the series given can be expressed as the sum of two infinite series (\( S_1 \) and \( S_2 \)), and both are geometric in nature.
Key points about infinite series include:

• They are expressed as sums continuing endlessly.
• Determining convergence is crucial as it dictates if a finite sum exists.

The concept of convergence plays a crucial role in the study of infinite series, helping to ascertain whether calculations like the one performed in the exercise will result in a meaningful value.