Problem 67 - Precalculus: Mathematics for Calculus

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots$$

Step-by-Step Solution

Answer

The series is convergent, and its sum is \( \frac{3}{4} \).

Step 1: Identify the First Term and Common Ratio

The first term of the series is the first term of the sequence, which is 1. To find the common ratio \( r \), divide the second term by the first term. Here, the second term is \(-\frac{1}{3}\), and the first term is 1, so the common ratio \( r = \frac{-1/3}{1} = -\frac{1}{3}\). The sequence is: \( 1, -\frac{1}{3},\frac{1}{9}, -\frac{1}{27}, \ldots \).

Step 2: Determine Convergence or Divergence

An infinite geometric series converges if the absolute value of the common ratio \( |r| < 1 \). Here, \(|r| = |-\frac{1}{3}| = \frac{1}{3}\), which is less than 1. Therefore, the series is convergent.

Step 3: Calculate the Sum of the Convergent Series

The sum \( S \) of a convergent infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Substituting \( a = 1 \) and \( r = -\frac{1}{3} \) into the formula gives: \[ S = \frac{1}{1 - (-\frac{1}{3})} = \frac{1}{1 + \frac{1}{3}} = \frac{1}{\frac{4}{3}} = \frac{3}{4}. \]

Key Concepts

Understanding Convergence in Infinite Geometric Series

Convergence is a key concept when dealing with infinite geometric series. When we talk about convergence, we refer to whether a series approaches a certain finite number as you add more and more terms. For a geometrical series, convergence depends on the common ratio, generally denoted as \( r \). To determine convergence:

• The series will converge if the absolute value of its common ratio \( |r| \) is less than 1.
• If \( |r| \geq 1 \), the series diverges, meaning it does not settle towards any particular value.

In our exercise, we have a common ratio of \( -\frac{1}{3} \). Since the absolute value, \( \left| -\frac{1}{3} \right| = \frac{1}{3} \), is less than 1, this series converges. This rule allows us to conclude that we can find a finite sum for this infinite series.

Exploring the Common Ratio

In a geometric series, the common ratio \( r \) is fundamental. It indicates how each term in the series is generated from the previous one. It is crucial in determining the nature of the series—whether it converges or diverges. To find the common ratio:

• Take the second term of the series and divide it by the first term.
• This formula is simple: \( r = \frac{{ ext{second term}}}{{ ext{first term}}} \).

In the example provided, the second term is \(-\frac{1}{3}\) and the first term is \(1\). Therefore, the calculation for the common ratio is\[ r = \frac{-\frac{1}{3}}{1} = -\frac{1}{3}. \]This negative ratio indicates that the series will alternate in sign, moving between positive and negative, but because its absolute value is less than 1, the series is convergent.

Calculating the Geometric Series Sum

Once we determine an infinite geometric series is convergent, we can calculate its sum using a specific formula. The sum of an infinite geometric series is given by the formula:\[ S = \frac{a}{1 - r} \]where \( S \) is the sum, \( a \) is the first term, and \( r \) is the common ratio. For our series:

• The first term \( a \) is \(1\).
• The common ratio \( r \) is \(-\frac{1}{3}\).

Plugging these values into the formula provides:\[ S = \frac{1}{1 - (-\frac{1}{3})} = \frac{1}{1 + \frac{1}{3}} = \frac{1}{\frac{4}{3}} = \frac{3}{4}. \]Thus, the sum of this specific infinite geometric series is \( \frac{3}{4} \). This result tells us that, although there are infinitely many terms, they combine to approach a finite number due to the properties of convergence and the specific common ratio.