Problem 57 - Precalculus Mathematics for Calculus

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 converges; its sum is 2.

Step 1: Identify the First Term and Common Ratio

Identify the first term of the series (\( a_1 \)) and the common ratio (\( r \)).The first term, \( a_1 \), is 3.To find the common ratio, \( r \), divide the second term by the first term: \(-\frac{3}{2} \div 3 = -\frac{3}{2} \times \frac{1}{3} = -\frac{1}{2}\).

Step 2: Determine Convergence or Divergence

For an infinite geometric series \( a_1 + a_1r + a_1r^2 + \cdots \), it is convergent if \( |r| < 1 \).In this series, \( r = -\frac{1}{2} \), and \(|r| = \frac{1}{2} < 1 \).Therefore, the series is convergent.

Step 3: Calculate the Sum of the Series

Since the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a_1}{1 - r} \]Plug in the values: \( a_1 = 3 \) and \( r = -\frac{1}{2} \):\( S = \frac{3}{1 - (-\frac{1}{2})} = \frac{3}{1 + \frac{1}{2}} = \frac{3}{\frac{3}{2}} = 2 \).Thus, the sum of the series is 2.

Key Concepts

Infinite Series

An infinite series is a sum of an infinite number of terms. Unlike a finite series, which comes to an end, an infinite series continues indefinitely. In mathematical notation, it's often represented as:

• \[ a + ar + ar^2 + ar^3 + \cdots \]

Here, \( a \) is the first term, and \( r \) is the common ratio between successive terms. The series involves adding more and more terms forever. A key question with infinite series is whether they sum to a finite value or not.
In the context of geometric series, whether the series actually converges (leads to a single finite sum) depends on the value of the common ratio, \( r \). Understanding this behavior is critical because it tells us whether we can meaningfully add up all the infinite terms. When studying infinite series, the idea is to determine if there is some point where the total stops growing indefinitely, and instead stabilizes around a particular number.

Convergence

Convergence in the context of series is a fundamental concept. It refers to the condition under which an infinite series approaches a fixed sum. For a geometric series, this happens if the absolute value of the common ratio \( r \) is less than 1. If \( |r| < 1 \), the series will converge, meaning that as you continue adding more terms, the total moves closer and closer to a certain number.

• If \( |r| \geq 1 \), the series is divergent and does not settle on any number.
• For example, in the series \( 3 - \frac{3}{2} + \frac{3}{4} - \frac{3}{8} + \cdots \), the common ratio is \( -\frac{1}{2} \). Because \( |-\frac{1}{2}| = \frac{1}{2} \), which is less than 1, the series converges.

Convergence is crucial because it indicates that an infinite series has a well-defined sum despite having infinitely many terms. It is like saying that after an infinite process, we can still end up with a finite result.

Sum of Series

Once it's established that a series converges, the next step is finding its sum. For an infinite geometric series with a first term \( a_1 \) and common ratio \( r \), the sum is given by the formula:

• \[ S = \frac{a_1}{1 - r} \]

This formula arises from the fact that as the terms of the series progress, the total approaches a certain sum. It’s a beautiful result that even an infinite process can have such a neat and tidy ending. To apply the formula, simply plug in the values for \( a_1 \) and \( r \).For the series in question, \( a_1 = 3 \) and \( r = -\frac{1}{2} \).
Therefore, the sum is calculated as:

• \[ S = \frac{3}{1 + \frac{1}{2}} = 2 \]

This reveals that despite the infinite number of terms, the series sums up to 2. Understanding the sum of a series helps quantify and make sense of infinite processes in a manageable way.