Problem 28 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The series is convergent with a sum of \( \frac{3}{8} \).

Step 1: Identify the Pattern

Examine the given series to identify a pattern. The terms appear to alternate between fractions with numerators 1 and 2. The denominators form a sequence: 3, 9, 27, 81, 243, 729, which suggests terms of a geometric series. Specifically, each denominator is a power of 3. Notice that these terms can be expressed as: \( \frac{1}{3^n} \) when the numerator is 1, and \( \frac{2}{3^{n+1}} \) when the numerator is 2.

Step 2: Express Series as Sums

Split the series into two separate series based on the pattern of numerators: \( \sum \frac{1}{3^n}, \ n=1,3,5,... \) and \( \sum \frac{2}{3^{n+1}}, \ n=2,4,6,... \). These are geometric series.

Step 3: Determine the Geometric Series Formula

Recall that a geometric series \( \sum a r^n \) converges if \(|r| < 1\) and its sum is \( \frac{a}{1-r} \). For both parts, the series first terms and common ratios will be identified.

Step 4: Analyze First Series

The series \( \sum \frac{1}{3^n}, n=1,3,5,... \) starts at \( n=1 \) with the term \( \frac{1}{3}\). The series can be rewritten as \( \frac{1/3}{1-(1/9)} \) because the common ratio, when considering only odd terms, is \( \frac{1}{9} \). The sum is \( \frac{1/3}{1-1/9} = \frac{1/3}{8/9} = \frac{1}{8} \).

Step 5: Analyze Second Series

For \( \sum \frac{2}{3^{n+1}}, n=2,4,6,...\), the first term is \( \frac{2}{9} \) with a common ratio \( \frac{1}{9} \). Its sum is \( \frac{2/9}{1-(1/9)} = \frac{2/9}{8/9} = \frac{1}{4} \).

Step 6: Combine Results

Add the sums from the two series: \( \frac{1}{8} + \frac{1}{4} = \frac{1}{8} + \frac{2}{8} = \frac{3}{8} \). Therefore, the total sum of the original series is \( \frac{3}{8} \).

Key Concepts

Geometric Series

A geometric series is a fascinating type of series where each term is a constant multiple, known as the common ratio, of the previous term. This consistent pattern makes geometric series easy to identify. For example, in a geometric series like \( a, ar, ar^2, ar^3, \ldots \), the common ratio \( r \) is multiplied with each preceding term.
One of the remarkable traits of geometric series is how simple they make calculations and predictions, thanks to their repetitive nature.

• When \( |r| < 1 \), the geometric series will converge, meaning it approaches a specific sum.
• If \(|r| \geq 1\), the series diverges, so it does not settle on any particular value.

To find the convergence, simply check the absolute value of \( r \). If it's less than 1, you're on a path toward convergence. It’s like walking down a hallway that eventually tapers off, funneling you toward a single exit point.

Sum of Series

Finding the sum of a geometric series, assuming it converges, is not as daunting as it may seem, thanks to a handy formula. For a geometric series with a first term \( a \) and common ratio \( r \), the sum is calculated with:\[S = \frac{a}{1 - r}\]
Here’s how it works in action: For the first series in the exercise, \( \frac{1}{3} + \frac{1}{27} + \cdots \), the first term \( a \) is \( \frac{1}{3} \) and the common ratio \( r \) is \( \frac{1}{9} \). Thus, the sum becomes \( \frac{1/3}{1 - 1/9} \). Calculations break this down to \( \frac{1}{8} \).
The magic of this formula lies in its ability to sum up an infinite number of terms into a single number, provided the series converges.

• This formula saves you from the tedious task of adding potentially countless terms.
• It turns what seems to be an impossible task of summing infinity, into a finitely quantified result.

Series Divergence

Series divergence is a concept that helps us understand what happens when a series doesn’t converge, meaning it doesn’t settle into a fixed sum. When a series is divergent, it can increase without bound or fluctuate indefinitely.For geometric series, diversification happens when the absolute value of the common ratio \( |r| \) is equal to or greater than 1. In simpler terms, divergence in a geometric series occurs when:

• Each term doesn't reliably shrink toward zero, preventing a limit from forming.
• The series just keeps growing, like climbing an infinite stairway.

Encountering a divergent series signals that achieving a stable sum is not possible, and acts as a crucial checkpoint in understanding whether to expect a meaningful numerical result. Recognizing patterns of divergence takes practice, but provides great insight into the behavior of infinite series.