Problem 4 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty}\left(-\frac{3}{2}\right)^{n} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the First Term and Common Ratio

The given series is \ \ \ \ \[ \sum_{n=0}^{\infty}\left(-\frac{3}{2}\right)^{n} \] \ \ In a geometric series, the expression can be written as \ \[ \sum_{n=0}^{\infty} ar^n \] \ \ where \( a \) is the first term and \( r \) is the common ratio. Here, the first term \( a \) is 1 (when \( n = 0 \)) and the common ratio \( r \) is \( -\frac{3}{2} \).

Step 2: Determine Convergence or Divergence

For a geometric series to converge, the absolute value of the common ratio \( |r| \) must be less than 1. Check whether \( |r| \) is less than 1: \[ |r| = \left|-\frac{3}{2}\right| = \frac{3}{2} > 1\] Since \( \frac{3}{2} \) is greater than 1, the series diverges.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the series \(\text{1, 2, 4, 8, 16}...\), each term is multiplied by 2 to get the next term. Geometric series can be written in summation notation like this: \[ \sum_{n=0}^{\infty} a r^n \] where \( a \) is the first term and \( r \) is the common ratio. Understanding how geometric series works allows us to determine important properties about the series, such as whether it converges or diverges.

Convergence and Divergence

When we say a geometric series converges, it means that the sum of all its terms approaches a finite value. On the other hand, if a series diverges, the sum keeps growing indefinitely and never settles to a finite number. Determining whether a geometric series converges or diverges depends on the absolute value of the common ratio, \(r\). For a geometric series to converge, the condition is that the absolute value of \( r \) must be less than 1. Mathematically, this is expressed as: \[ |r| < 1 \] If \( |r| \) is greater than or equal to 1, the series diverges. In our example series, given by \[ \sum_{n=0}^{\infty}\left(-\frac{3}{2}\right)^{n} \], the common ratio \( r \) is \( -\frac{3}{2} \). Since \( \left| -\frac{3}{2} \right| = \frac{3}{2} > 1 \), the series diverges.

Common Ratio

The common ratio, denoted as \(r\), is the factor by which each term in a geometric series is multiplied to get the next term. It plays a crucial role in determining the behavior of the series. Let's break down how to find and interpret the common ratio in any given geometric series. For instance, in the series \[ \sum_{n=0}^{\infty}\left(-\frac{3}{2}\right)^{n} \], if we look closer, we see the common ratio \( r \) is \( -\frac{3}{2} \). We calculate the absolute value to decide on the convergence as \[ |r| = \left| -\frac{3}{2} \right| = \frac{3}{2} \]. Since this value is greater than 1, the series does not settle to a finite sum, meaning it diverges. Remember, checking whether \( |r| \) is less than 1 is a reliable method for determining the nature of a geometric series. If \( r \) falls within this range, you can also find the sum of the series through the formula \[ \text{Sum} = \frac{a}{1-r} \] where \( a \) is the first term.