Problem 10 - Single Variable Calculus

Question

Determine whether the geometric series converges or diverges. If it converges, find its sum. \(1+\frac{4}{3}+\frac{16}{9}+\frac{64}{27}+\cdots\)

Step-by-Step Solution

Answer

The geometric series diverges because the absolute value of the common ratio, \(|r| = \frac{4}{3}\), is greater than 1. Therefore, the sum does not exist for this series.

Step 1: Identify the common ratio

Observe the given geometric series and determine the common ratio (r) by dividing any term by the previous term: \[\frac{\frac{4}{3}}{1} = \frac{4}{3}\]

Step 2: Check for convergence

Calculate the absolute value of the common ratio to determine whether the series converges or diverges. If \(|r| < 1\), the series converges, otherwise it diverges. \[|r| = \left|\frac{4}{3}\right| = \frac{4}{3}\] Since \(\frac{4}{3} > 1\), the geometric series diverges. As a result, there is no sum for the geometric series, as it diverges.

Key Concepts

Common Ratio

In any geometric series, each term is obtained by multiplying the previous term by a constant value, known as the common ratio. Understanding this ratio is crucial to analyzing the behavior of the series. To find the common ratio, you simply divide one term in the series by the term immediately before it. In our example, the second term is \(\frac{4}{3}\) and the first term is \(1\), so the common ratio \(r\) is:

• \(r = \frac{\frac{4}{3}}{1} = \frac{4}{3}\)

It's important to note that the common ratio can be positive or negative and even a fraction. Determining this common factor helps decide whether the series will grow, shrink, or oscillate.

Convergence

Geometric series can either converge or diverge depending on the absolute value of the common ratio. Convergence occurs when the series approaches a particular value as more terms are added. This happens if the absolute value of the common ratio is less than 1. In mathematical terms:

• If \(|r| < 1\), the series converges.

Converging series are interesting because they allow us to find a finite sum, even though the series might have infinitely many terms. Unfortunately, in our example, the common ratio is \(\frac{4}{3}\), which is greater than 1. Thus, the series does not converge.

Divergence

A geometric series diverges when it doesn't settle around any finite value as you add more terms. This typically happens when the absolute value of the common ratio is greater than or equal to 1:

• If \(|r| \geq 1\), the series diverges.

In our example, since the common ratio \(r = \frac{4}{3}\) exceeds 1, the series keeps growing larger, never close to a particular limit. Divergence implies that we cannot calculate a sum for the series since the terms will keep increasing indefinitely, making it unable to converge to a specific value.

Sum of Series

When a geometric series converges, it's possible to calculate its sum. The formula used for finding the sum \(S\) of a convergent geometric series with first term \(a\) and common ratio \(r\) is:\[S = \frac{a}{1-r}\]This formula only applies if \(|r| < 1\). However, since in our scenario the common ratio is \(\frac{4}{3}\), which makes the series diverge, this formula cannot be used. Therefore, the sum does not exist for this specific series, since a divergent series doesn't have a finite total. Always ensure that the common ratio's absolute value is below 1 before applying this sum formula.