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. $$ \frac{\text { Function }}{f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}} $$

Short answer

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

Step-by-step solution

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.

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.

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.

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.