Problem 111 - Precalculus with Limits

Question

In Exercises 111 and 112, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. \( f(x) = 6\left[\dfrac{1 - \left(0.5\right)^x}{1 - \left(0.5\right)}\right] \), \( \sum_{n=0}^{\infty}6\left(\frac{1}{2}\right)^n \)

Step-by-Step Solution

Answer

The horizontal asymptote of the given function is \(y = 12\) and the sum of the given series is also 12. The relationship between them is that the sum of the infinite geometric series is the horizontal asymptote of the function that represents its partial sums.

Step 1: Graphing the Function and Identifying the Asymptote

Graph the function \( f(x) = 6\left[\dfrac{1 - \left(0.5\right)^x}{1 - \left(0.5\right)}\right] \) using a graphing utility and identify the horizontal asymptote. It is observed that the curve approaches a horizontal line as x tends towards infinity which proves that this curve has a horizontal asymptote, which is \( y = 12 \). This means as \( x \) moves towards infinity, \( f(x) \) approaches 12.

Step 2: Computing the Sum of Geometric Series

Now compute the sum of the given geometric series \( \sum_{n=0}^{\infty}6\left(\frac{1}{2}\right)^n \). The sum S of an infinite geometric series, where |r| < 1, is given by the formula \( S = \dfrac{a}{1 - r} \), where a is the first term and r is the common ratio. The first term \( a = 6 \) and the common ratio \( r = 0.5 \). So, \( S = \dfrac{6}{1 - 0.5} = 12 \).

Step 3: Comparing the Sum with the Asymptote

Observe that the sum of the series and the horizontal asymptote are same that is 12. This confirms that this function represents the partial sums of this geometric series, where the sum of the infinite geometric series is the horizontal asymptote of the function.

Key Concepts

Horizontal Asymptote

When we talk about a horizontal asymptote, we refer to a line that a graph approaches as the input values get larger or smaller indefinitely. For the function given, \( f(x) = 6\left[\dfrac{1 - \left(0.5\right)^x}{1 - \left(0.5\right)}\right] \), the horizontal asymptote is confirmed to be \( y = 12 \).
This implies that as \( x \) continues approaching infinity, the value of \( f(x) \) gets closer and closer to 12, but never actually reaches it.

• A horizontal asymptote doesn't mean the function can't cross that line—it's just where it levels out as \( x \) becomes very large or small.
• Understanding horizontal asymptotes helps to predict the long-term behavior of a function.

By identifying that \( y = 12 \) is the horizontal asymptote, we learn about how the function behaves at extreme values of \( x \). This is crucial when analyzing infinite sequences and series.

Graphing Utility

Using a graphing utility plays a vital role in visualizing mathematical functions and their characteristics. For instance, let's consider how it helps in analyzing the given function \( f(x) = 6\left[\dfrac{1 - \left(0.5\right)^x}{1 - \left(0.5\right)}\right] \). By inputting the function into a graphing utility, we can quickly see how the graph behaves.
With a graphing utility, you can:

• Observe how the curve of the function approaches its horizontal asymptote as \( x \) increases.
• Check the symmetry, intercepts, and end-behaviors of the function.

Such tools provide instant visual insights, making complex mathematical concepts more tangible and easier to understand. By using them, students can experiment and learn how certain functions and their parameters affect their graphs.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. In this exercise, we focus on the infinite geometric series \( \sum_{n=0}^{\infty}6\left(\frac{1}{2}\right)^n \). Let's break it down to understand it better.
Key Characteristics of the Infinite Series:

• It has a first term (\( a = 6 \)) and a common ratio (\( r = 0.5 \)).
• The common ratio is essential to determine the behavior of the series. Here, \( |r| < 1 \) indicates that the series converges.

The sum of this infinite series is found using the formula \( S = \dfrac{a}{1 - r} \), which gives us a total of 12. In summary, an infinite series expands endlessly, but under certain conditions, its total can neatly sum up to a finite value.

Convergence of Series

The concept of convergence is crucial when dealing with infinite series. A series is considered to converge if the sum of its terms approaches a specific finite number as the number of terms increases indefinitely.
In our function's associated series, \( \sum_{n=0}^{\infty}6\left(\frac{1}{2}\right)^n \), the series converges to 12.
These aspects lead to convergence:

• The common ratio \( r \) must satisfy \( |r| < 1 \).
• The calculation of the sum is made possible through the convergence formula \( \dfrac{a}{1-r} \).

Understanding the convergence of a series helps in predicting long-term trends and overall behavior of more complex functions involving infinite sums. Hence, it gives meaningful predictions and solutions to intricate mathematical problems.