Problem 53 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the First Term and Common Ratio

Let's examine the given series: \(1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \cdots\). The first term \(a\) is \(1\). The common ratio \(r\) is obtained by dividing the second term \(-\frac{1}{3}\) by the first term \(1\), i.e. \(r = -\frac{1}{3}\).

Step 2: Determine Convergence/Divergence

For an infinite geometric series to converge, the common ratio \(r\) must satisfy \(|r| < 1\). Here, \(|r| = |-\frac{1}{3}| = \frac{1}{3} < 1\). Thus, the series is convergent.

Step 3: Calculate the Sum of the 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. Substitute \(a = 1\) and \(r = -\frac{1}{3}\) into the formula: \[ S = \frac{1}{1 - (-\frac{1}{3})} = \frac{1}{1 + \frac{1}{3}} = \frac{1}{\frac{4}{3}} = \frac{3}{4}. \]

Key Concepts

Convergence

In an infinite geometric series, convergence is a vital concept that helps us determine if the series has a finite sum. An infinite series converges if the sum of its terms approaches a specific value as the number of terms grows indefinitely. For a geometric series,

• The convergence depends on its common ratio, often denoted as \( r \).
• The series converges if the absolute value of this ratio is less than 1, i.e., \(|r| < 1\).

If the series does not converge, it is said to diverge, meaning the series will not settle on a finite value. In our exercise, the given series is \(1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \cdots\). Calculating the common ratio, we find it is \(-\frac{1}{3}\), with an absolute value of \(\frac{1}{3}\). Since \(\frac{1}{3} < 1\), the series converges, indicating that it possesses a finite sum.

Common Ratio

The common ratio \( r \) is essential for understanding and evaluating the behavior of a geometric series. It is the factor by which each term in the series is multiplied to get the next term. Identifying this ratio involves dividing any term by the preceding term in the series. In our example,

• The first term (\( a \)) is 1.
• The second term is \(-\frac{1}{3}\).
• Thus, the common ratio \( r \) is \(-\frac{1}{3} \div 1 = -\frac{1}{3}\).

The common ratio not only tells us how the series behaves but also plays a crucial role in determining convergence. In our scenario, the fact that \(|r| = \frac{1}{3} < 1\) means the series will converge. By understanding the common ratio, we gain insight into how fast the series terms decrease and whether the series converges or diverges.

Sum of Series

The sum of an infinite geometric series is fascinatingly simple to compute when the series converges. For a convergent series with a common ratio \(|r| < 1\), the sum \( S \) can be calculated using the formula: \[S = \frac{a}{1 - r}.\] Here, \( a \) is the first term, and \( r \) is the common ratio. Let's apply this to our series:

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

Substituting these values into the formula gives:\[S = \frac{1}{1 - (-\frac{1}{3})} = \frac{1}{1 + \frac{1}{3}} = \frac{1}{\frac{4}{3}} = \frac{3}{4}.\] This calculation reveals that the sum of our series is \(\frac{3}{4}\), demonstrating the power of understanding and applying the formula for the sum of a geometric series. This approach not only provides the sum but also underscores the patterns found in infinite series.