Problem 13 - Algebra 2

Question

Decide whether each infinite geometric series diverges or converges. State whether each series has a sum. $$ 6+18+54+\ldots $$

Step-by-Step Solution

Answer

The infinite geometric series diverges, and thus does not have a sum.

Step 1: Identify the Common Ratio

To identify the common ratio of the geometric series, we divide the second term by the first term, or any term by its preceding term. Thus the common ratio here is \( \frac{18}{6} = 3 \) .

Step 2: Check Conditions for Convergence

A geometric series converges if the absolute value of its common ratio is less than 1. In other words, \( |r| < 1 \). Since our common ratio is 3, which is greater than 1, it follows that the series diverges.

Step 3: Find Sum of Series

The geometric series sum formula is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. However, this is only applicable when \( |r| < 1 \) and hence the series converges. In this case since our series diverges, it doesn't have a sum.

Key Concepts

Convergence and Divergence

When we talk about infinite geometric series, understanding whether a series converges or diverges is crucial. A geometric series converges if the absolute value of its common ratio, denoted as \(|r|\), is less than 1. In simple terms, this condition ensures that the terms in the series get smaller and closer to zero as the series extends infinitely. If this happens, the sum of the infinite series reaches a finite value.

On the other hand, if the absolute value of the common ratio is equal to or greater than 1, the series diverges. This implies that the terms do not diminish as you progress through the series, hence there is no finite sum.

For our series, \(6 + 18 + 54 + \ldots\), the common ratio is 3, which is much greater than 1. Therefore, this series diverges, and its sum is not finite.

Common Ratio

The common ratio is an essential element in a geometric series. It is the factor by which each term in the series is multiplied to get the next term. To find the common ratio, divide any term in the series by the preceding one. For instance, in our series \(6 + 18 + 54 + \ldots\), the second term is 18 and the first term is 6. Thus, the common ratio \(r\) is calculated as \( \frac{18}{6} = 3 \).

It's important to mention that once you have the common ratio, it tells you a lot about the behavior of the series:

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

In our example, since \(r = 3\), the series diverges, confirming that it does not have a finite sum.

Geometric Series Sum Formula

The geometric series sum formula is a powerful tool when dealing with converging geometric series. The formula is given by \( S = \frac{a}{1 - r} \), where \(a\) is the first term of the series and \(r\) is the common ratio. This formula calculates the sum of all the terms in a convergent infinite geometric series.

However, this formula is only valid under the condition that the absolute value of the common ratio \(|r|\) is less than 1. If this condition is not met, the series diverges and the formula cannot be used since the series does not have a finite sum.

In our series example, where \(r = 3\), the formula is not applicable as the series diverges, and hence, there is no sum to calculate. Understanding this limitation is crucial when working with geometric series.