Problem 57 - Algebra and Trigonometry

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\cdots $$

Step-by-Step Solution

Answer

The series is convergent, and its sum is 2.

Step 1: Identify the First Term and Common Ratio

For this geometric series, the first term is \( a = 3 \). To find the common ratio \( r \), divide the second term by the first term: \(-\frac{3}{2} \div 3 = -\frac{1}{2}\). Thus, the common ratio is \( r = -\frac{1}{2} \).

Step 2: Determine Convergence or Divergence

A geometric series converges if the absolute value of the common ratio is less than 1, i.e., \(|r| < 1\). In this series, \(|r| = \left| -\frac{1}{2} \right| = \frac{1}{2} \), which is less than 1. Thus, the series is convergent.

Step 3: Calculate the Sum of the Convergent Series

The sum \( S \) of an infinite convergent geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Plug in the values: \( S = \frac{3}{1 -(-\frac{1}{2})} = \frac{3}{1 + \frac{1}{2}} = \frac{3}{\frac{3}{2}} = 2 \). Thus, the sum of the series is 2.

Key Concepts

Convergent Series

An infinite series is said to be convergent if the sum of its terms approaches a certain finite number as the number of terms increases. In the case of an infinite geometric series, convergence depends largely on the common ratio. If the absolute value of the common ratio, denoted as \(|r|\), is less than 1, the series will converge. This means that even though you're adding an infinite number of terms together, these terms get smaller and smaller, allowing the sum to settle at a specific finite value.
For the geometric series described in the original exercise, we found that the common ratio \(r\) is \(-\frac{1}{2}\), which makes \(|r| = \frac{1}{2}\). Since \(\frac{1}{2} < 1\), this series is convergent. This is a crucial assessment because it tells us that we can now find a total finite sum for this series.

Common Ratio

The common ratio in a geometric series is the factor by which you multiply each term to get to the next term. It is a constant that remains the same throughout the series. The common ratio is calculated by dividing any term in the series by the preceding term.

• If the common ratio \(|r| < 1\), the series is convergent.
• If \(|r| \geq 1\), the series is divergent and no finite sum exists.

In the exercise example, the first term \(a\) is 3. To find the common ratio, we divided the second term \(-\frac{3}{2}\) by the first term, which gave us the common ratio \(r = -\frac{1}{2}\). This specific value shows us the factor which consistently affects how each term is calculated from the one before it, significantly impacting the behavior of the series.

Geometric Series Sum Formula

When a geometric series is convergent, we can find its sum using a specific formula. The sum \(S\) of an infinite convergent geometric series is given by:\[S = \frac{a}{1 - r}\]Here, \(a\) represents the first term of the series, and \(r\) is the common ratio. This formula works only if the series is convergent, meaning \(|r| < 1\).
For the series in question, we apply this formula with \(a = 3\) and \(r = -\frac{1}{2}\). By substituting these values into the formula, we get:\[S = \frac{3}{1 - (-\frac{1}{2})} = \frac{3}{1 + \frac{1}{2}} = \frac{3}{\frac{3}{2}} = 2\]Thus, the sum of the series is 2. This result illustrates not only how easily such series can be summed when they converge but also how vital understanding the behavior of the common ratio is.