Problem 24 - Algebra 2

Question

Evaluate each infinite geometric series. $$ 3-2+\frac{4}{3}-\frac{8}{9}+\dots $$

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is \(S =9/5\).

Step 1: Identify the First Term (a)

The first term in any series is simply the first number listed. In this case, the first term \(a\) is 3.

Step 2: Identify the Common Ratio (r)

The common ratio \(r\) in a geometric sequence is the constant ratio between two successive terms. We can find it by dividing the second term by the first term. So in this case, we have \(r = -2 / 3 = -2/3.\)

Step 3: Test For Convergence

Before we can use the formula to find the sum of an infinite series, we need to ensure that it does indeed converge. An infinite geometric series \(S = a / (1 - r)\) will only converge if the absolute value of \(r\) is less than 1. In this case, the absolute value of -2/3 is \(|-2/3| = 2/3\) which is less than 1. Therefore, the series does converge.

Step 4: Find the Sum of the Series

Since the series converges, we can use the formula for the sum of an infinite geometric series \(S = a / (1 - r)\) to find the sum. Plugging in \(a = 3\) and \(r = -2/3\), we get \(S = 3 / (1 - (-2/3)) = 3 / (3/3 + 2/3) = 3 / (5/3) = 3 * (3/5) = 9/5\).

Key Concepts

Geometric Sequence

A geometric sequence is an ordered set of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This particular sequence:

• Begins with the number 3.
• The next terms follow this principle of multiplying by a set ratio.

In the series given in the exercise, each term is clearly generated by a continued multiplication process, showcasing the basic characteristic of a geometric sequence. Understanding how each term is found is crucial in identifying the larger pattern or rule that the sequence follows. This will be useful when trying to evaluate the series as a whole, particularly when the sequence continues indefinitely.

First Term

In any geometric sequence or series, the first term is denoted by the symbol \(a\). It serves as the starting point for the sequence.

• In our exercise, the first term \(a\) is 3.
• This first term plays a critical role in calculating the sum of the series, as it is used in the sum formula for geometric sequences.

Always make sure to clearly identify the first term before proceeding with any calculations, as this ensures every subsequent term and the overall series calculation is accurate. Highlighting the first term helps in determining whether the series will converge and thus be evaluable for further operations.

Common Ratio

The common ratio \(r\) is what defines the continuous multiplication pattern of a geometric sequence. It's the factor between any two successive terms and can be calculated by dividing any term by its preceding term. In this scenario:

• The common ratio between 3 and -2 is \(-\frac{2}{3}\).
• Being a constant, this ratio sets the pace for the growth or decay of the sequence.

This ratio is crucial for analyzing the behavior of the series over time, especially in infinite series where convergence depends on this value. Understanding the common ratio allows you to anticipate how the sequence progresses and ensures you can accurately compute the sum of the series when possible.

Convergence

Convergence in infinite series is a core concept when evaluating their sums. A series converges if the sequence of partial sums approaches a finite limit. For geometric series, this occurs when the absolute value of the common ratio \(r\) is less than 1.

• In our example, the common ratio \(-\frac{2}{3}\) satisfies this condition, as its absolute value is \(\frac{2}{3}\), which is less than 1.
• This means we can calculate the sum of the infinite series using the formula \(S = \frac{a}{1-r}\).

Convergence indicates that although the series has infinitely many terms, their sum approaches a finite value. This is both a fascinating and practical property when examining infinite series, providing results that can be used in various mathematical, physical, or economical applications.