Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Find the sum of the infinite geometric series if it exists. $$1+\frac{3}{2}+\frac{9}{4}+\frac{27}{8}+\cdots$$

Step-by-Step Solution

Answer

The series does not converge, so the sum does not exist.

Step 1: Identify First Term and Common Ratio

To find the sum of an infinite geometric series, first identify the first term and any common ratio. The series given is \(1 + \frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \cdots\). Here, the first term \(a\) is 1. To find the common ratio \(r\), divide the second term by the first term: \(\frac{3}{2} \div 1 = \frac{3}{2}\). Verify by checking a few terms: \(\frac{9}{4} \div \frac{3}{2} = \frac{3}{2}\) and \(\frac{27}{8} \div \frac{9}{4} = \frac{3}{2}\). The common ratio \(r\) is \(\frac{3}{2}\).

Step 2: Check for Convergence

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. In this case, \(|\frac{3}{2}| = 1.5\), which is greater than 1. Hence, the series does not converge.

Step 3: Conclusion on the Sum

Since \(|r| > 1\), the infinite geometric series does not have a sum. Therefore, the sum of this infinite geometric series does not exist.

Key Concepts

Geometric Series Convergence

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. Understanding *convergence* is crucial to determine if we can find a sum for an infinite geometric series. An infinite geometric series will converge, or approach a finite sum, if and only if the absolute value of the common ratio, denoted as \(|r|\), is less than 1. This means the terms decrease in magnitude, getting closer and closer to zero, allowing the series to sum up to a specific number.

• If \(|r| < 1\), the series converges, and the sum can be calculated.
• If \(|r| \geq 1\), the series diverges, meaning it does not sum to a finite number.

In our exercise example, the common ratio \(r = \frac{3}{2} = 1.5\), which is greater than 1. This tells us immediately that the series does not converge, as the sequence of terms actually increases in magnitude. Hence, the series does not have a sum.

Common Ratio

The common ratio is a key element in understanding geometric series. It is found by taking any term in the series and dividing it by the preceding term. Consistency is crucial here; the common ratio should be the same between any two consecutive terms for the series to be truly geometric.For example, in the series given in the exercise: \(1 + \frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \cdots\), we calculate it as:

• \( \frac{3}{2} \div 1 = \frac{3}{2} \)
• \( \frac{9}{4} \div \frac{3}{2} = \frac{3}{2} \)
• \( \frac{27}{8} \div \frac{9}{4} = \frac{3}{2} \)

The common ratio of \( \frac{3}{2} \) is constant throughout. Identifying this ratio helps us determine the nature of the series, whether it converges or not, and ultimately if we can find its sum.

Sum of Geometric Series

Once the convergence of a geometric series is established, we can find the sum if it converges. The formula to find the sum \( S \) of an infinite geometric series is:\[S = \frac{a}{1-r}\] Where \( a \) is the first term and \( r \) is the common ratio. This formula only applies if \(|r| < 1\), ensuring convergence. The math works because the terms progressively become smaller and add up to a finite value.However, in the exercise's series, with a common ratio \( r = \frac{3}{2} \), which is more than 1, the series will not sum to a finite number. Therefore, the sum cannot be determined because the series does not converge.To summarize, finding the sum of a geometric series requires not only the identification of the first term and common ratio but also verifying that the common ratio’s absolute value is less than 1.