Problem 41 - Algebra 2

Question

Open-Ended Write an infinite geometric series that converges to \(3 .\) Use the formula to evaluate the series.

Step-by-Step Solution

Answer

The geometric series that converges to 3 and satisfies the conditions is: \(3, 1.5, 0.75, 0.375, ...\)

Step 1: Determine the Terms of the Series

Choose an initial term, \(a = 3\), and a common ratio, \(r=0.5\), which must be between -1 and 1 for convergence. The series then becomes \(3, 3*0.5, 3*0.5^2, 3*0.5^3, ...\)

Step 2: Double check the Convergence

Confirm that the absolute value of the common ratio is less than 1. In this case the common ratio is \(0.5\) and so it meets this rule.

Step 3: Evaluate the Series using the Formula

Plug the values into the formula to verify whether the series sums to 3: \[S = \frac{a}{1 - r} = \frac{3}{1 - 0.5} = 3\]

Key Concepts

Convergent Series

In mathematics, when we talk about a convergent series, we're referring to an infinite series whose terms approach a specific value as the number of terms increases. The series "converges" to this finite limit, meaning that regardless of how many terms are added, the sum will not exceed a certain number. This is particularly interesting for infinite geometric series where proportions remain constant, yet the total sum stabilizes.
Importantly, for a geometric series to be convergent, the common ratio, often denoted as \(r\), must satisfy a very particular condition. Specifically, the absolute value of \(r\) needs to be less than 1—\(|r| < 1\). If this holds true, the series converges; otherwise, the series diverges, continuing to grow without limit.
In the example, by choosing \(r = 0.5\), which is indeed less than 1, we assure convergence. This allows the total sum of the infinite sequence to settle at a finite number, in this case, 3.

Geometric Series Formula

The geometric series formula provides the foundation for determining the sum of a series where each term is a constant multiple of the previous one. It is articulated as:

• \(S = \frac{a}{1 - r}\)

Here, \(a\) is the first term of the series, and \(r\) is the common ratio.
This formula is pertinent for infinite geometric series which are convergent, specifically when the condition \(|r| < 1\) is met. By using this formula, we can straightforwardly calculate the total sum that the series will approach no matter how many terms are summed up.
In the context of the given exercise, the initial term \(a\) is 3, and the common ratio \(r\) is 0.5. Plugging these into the formula gives:
\[S = \frac{3}{1 - 0.5} = 3\]
This confirms that the sum of this infinite geometric series indeed converges to 3 as expected.

Common Ratio

When dealing with a geometric series, the common ratio is a crucial component. This ratio is the factor by which you multiply each term of the series to get the next term. For any geometric series defined by terms such as \(a, ar, ar^2, ar^3, ...\), \(r\) defines this relationship.
The value of the common ratio dictates whether a series will converge or diverge. If the absolute value of the common ratio \(|r|\) is less than 1, the series converges, meaning its sum will approach a fixed value.
Visually, in our problem, we choose \(r = 0.5\). This selection is intentional to ensure convergence, knowing that it satisfies \(|r| < 1\). Each successive term becomes smaller, making the sum approach a certain limit, neatly demonstrating the concept of an approaches-defined limit in infinite series.