Problem 88 - Essential Calculus

Question

Use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ \frac{\text { Function }}{f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function is 10, which is also the sum of the given series.

Step 1: Plot the function

Use a graphing utility to plot the function \(f(x) = 2 \cdot \frac{1 - (0.8)^x}{1 - 0.8}\). Look at the behavior of the function as x approaches positive and negative infinity. This will give you the horizontal asymptotes.

Step 2: Identify the horizontal asymptote

Using the graph, observe that as \(x\) tends to infinity, the function approaches a certain value. This value is the horizontal asymptote. You'll see that the function is progressively getting closer to a line (y = some_constant). That constant is the horizontal asymptote.

Step 3: Solve the infinite series

Evaluate the series \( \sum_{n=0}^{\infty} 2 \cdot \left(\frac{4}{5}\right)^n \) using the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(a\) is the first term (2 in this case), and \(r\) is the common ratio (4/5 here). Calculate the sum. The common ratio is less than 1, so it's possible to calculate the sum.

Step 4: Compare the sum and the asymptote

Compare the sum of the series obtained in Step 3 with the horizontal asymptote obtained in Step 2. You should find that the sum of the series and the horizontal asymptote are the same. This means that the horizontal asymptote of the function corresponds with the sum of the infinite series.

Key Concepts

Understanding Graphing Utility

A graphing utility is an invaluable tool that helps visualize mathematical functions. It can be software on a computer or a dedicated handheld device like a graphing calculator. When you input a function into this tool, it plots the graph for you, allowing you to observe its behavior quickly. This visualization helps you identify key features such as intercepts, vertices, and asymptotes.

In this exercise, using a graphing utility allows us to see how the function behaves as the input values (x) get very large or very small, which is crucial for identifying horizontal asymptotes. By graphing the function, you can easily determine where the function levels off, which provides a visual representation of the horizontal asymptote.

Exploring Geometric Series

A geometric series is a series of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). If the terms continue to infinity, the series is called an infinite geometric series.

The expression for the sum of an infinite geometric series is given by \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. If \( |r| < 1 \), the series converges, which means it approaches a fixed sum. If \( |r| \geq 1 \), the series does not converge.

In our context, we utilize this formula to find the sum of the series provided, which relates closely to the horizontal asymptote of our function.

Delving into Infinite Series

An infinite series is simply the sum of infinitely many numbers, which often appears in mathematical analysis and calculus. Series are foundational in defining functions not easily expressed in terms of finite algebraic terms.

An infinite series can be convergent or divergent. A convergent infinite series has a finite sum. Its partial sums get closer to a specific number as you add more terms. The infinite geometric series we have in this context is convergent because its common ratio is less than one.

Understanding how to manipulate and compute infinite series is crucial when analyzing functions, as their sum can equate to certain function behaviors such as horizontal asymptotes.

Calculating the Sum of the Series

The sum of an infinite geometric series can be calculated using the formula \( S = \frac{a}{1 - r} \). This calculation is essential as it gives us a precise number, which often has real-world applications, like in this exercise where it corresponds to the horizontal asymptote.

For the given series \( \sum_{n=0}^{\infty} 2 \left(\frac{4}{5}\right)^{n} \), we identify \( a = 2 \) and \( r = \frac{4}{5} \). Plugging these values into the formula, you find the sum is \( \frac{2}{1 - \frac{4}{5}} = 10 \). Doing this confirms that the sum of the series is the same as the horizontal asymptote identified from the graphing utility, illustrating a beautiful mathematical connection.