Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1} $$

Short answer

The series diverges since the common ratio is greater than 1.

Step-by-step solution

Identify the first term (a) and common ratio (r)

The given series is \ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1}.\ The first term (a) of the series is the value of the series when k = 1. Substitute k = 1: \[ a = 3\left(\frac{3}{2}\right)^{1-1} = 3\left(\frac{3}{2}\right)^{0} = 3 \]. The common ratio (r) is found by dividing the second term by the first term. Substitute k = 2 into the general term: \[ \text{Second term} = 3\left(\frac{3}{2}\right)^{2-1} = 3\left(\frac{3}{2}\right). \ \text{Common ratio} (r) = \frac{3\left(\frac{3}{2}\right)}{3} = \frac{3}{2} \].

Determine if the series converges

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1 (\|r\| < 1). Here, the common ratio is \( \frac{3}{2} \). Since \| \frac{3}{2} \| = \frac{3}{2} > 1, the series diverges.

Key concepts

converge

In the context of an infinite geometric series, convergence refers to whether the sum of the series approaches a finite value as more terms are added. For a series to converge, the absolute value of the common ratio, usually denoted by \(r\), must be less than 1. This means \|r| < 1\. When this condition is met, even though the number of terms grows infinitely, their sum converges to a specific value.
An example of a converging geometric series is: \3 + 3 \left(\frac{1}{2}\right) + 3\left(\frac{1}{2}\right)^2 + \ldots\
For this series, the common ratio \(r\) is \left(\frac{1}{2}\right)\. Since \| \frac{1}{2} | = \frac{1}{2}\, which is less than 1, the series converges. To find the sum of a converging series, you can use the formula: \[ S = \frac{a}{1-r} \] where \a\ is the first term and \r\ is the common ratio.

diverge

Divergence in an infinite geometric series means that the sum of the series does not approach a finite limit as more terms are added. Instead, the sum grows without bound. For divergence to occur, the absolute value of the common ratio \(r\) must be greater than or equal to 1. This means \|r| \geq 1\. A series where this happens is considered to diverge.
Let's consider the original exercise. The series is: \[ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1}\] Here, the common ratio \r\ is \frac{3}{2}\. Since \| \frac{3}{2} | = \frac{3}{2} > 1\, the series does not converge.
This means that as we keep adding terms, the sum will keep increasing and never settle at a finite number. Therefore, this series diverges.

common ratio

The common ratio is a crucial component in understanding whether a geometric series converges or diverges. It is the factor by which we multiply each term in the series to get the next term. If we denote the first term by \a\ and the second term by \ar\, then the common ratio \r\ is found by dividing the second term by the first term: \[ r = \frac{\text{Second Term}}{\text{First Term}} \]
Examining our series: \[ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1}\] we can identify:

• The first term \a = 3\
• The second term \a\left(\frac{3}{2}\right) = 3\left(\frac{3}{2}\right)\

We get the common ratio by dividing the second term by the first term: \[ r = \frac{3 \( \frac{3}{2} \)}{3} = \frac{3}{2} \]
A common ratio greater than 1, like in this example, implies the series will diverge.

first term

The first term of a geometric series, often denoted as \(a\), is where the sequence starts. It is critical to identifying the behavior of the series. Knowing the first term allows us to determine the future terms and even calculate the series' sum when applicable.
In our exercise, the first term is computed when \(k = 1\). For the series \[ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1}\] Upon substituting \( k=1 \), we get:
\[ a = 3\left(\frac{3}{2}\right)^{1-1} = 3\left(\frac{3}{2}\right)^0 = 3 \] Therefore, the first term \(a = 3\). This initial value helps set the pattern for the entire geometric series, and, combined with the common ratio, can help determine whether the series converges or diverges.