Problem 86 - Essential Calculus

Question

(a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1-\frac{x}{2}+\frac{x^{2}}{4}-\frac{x^{3}}{8}+\cdots $$

Step-by-Step Solution

Answer

The common ratio for the geometric series is \( -\frac{x}{2} \). The sum of series function is \( S = \frac{2}{2 + x} \). The graph of the function and the partial sums \( S_{3} \) and \( S_{5} \) shows convergence of the series with increasing number of terms.

Step 1: Find the Common Ratio

The common ratio \( r \) of a geometric series can be found by dividing any term in the series by the preceding term. Let's use the first two terms of the series. So, \( r = -\frac{x/2}{1} = -\frac{x}{2} \)

Step 2: Write the Sum Function

Now, with the common ratio and the first term of the series, the sum of the series can be written as \( S = \frac{a}{1 - r} \). Using the values \( a = 1 \) (the first term of the series) and \( r = -x/2 \), we substitute to get the sum equation which is \( S = \frac{1}{1 - (-x/2)} = \frac{1}{1 + \frac{x}{2}} = \frac{2}{2 + x} \).

Step 3: Graph the Function and the Partial Sums

Use a graphing tool to plot the function \( \frac{2}{2 + x} \) and the partial sums \( S_{3} \) and \( S_{5} \). \( S_{3} \) is the sum of the first three terms of the series and \( S_{5} \) is the sum of the first five terms. Calculate these partial sums from the series terms. Then plot \( S, S_{3}, \) and \( S_{5} \).

Key Concepts

Common Ratio

The common ratio in a geometric series is a fundamental aspect that allows us to understand the behavior of the sequence. To calculate it, we simply divide any term in the series by the preceding term. For example, considering the series given:

• The first term is 1.
• The second term is \(-\frac{x}{2}\).

To find the common ratio, take the second term and divide it by the first term, resulting in a ratio of \(-\frac{x}{2}\). This ratio is consistent throughout the series, meaning every term is obtained by multiplying the previous term by \(-\frac{x}{2}\). It’s essential to understand that this ratio determines the growth or decay pattern of the series.

Partial Sums

A partial sum of a series represents the sum of the initial segment of terms in the full series. In our context, partial sums like \(S_3\) and \(S_5\) enable us to examine how the series converges gradually towards the total sum.

• The partial sum \(S_3\) includes the first three terms: \(1 - \frac{x}{2} + \frac{x^2}{4}\).
• The partial sum \(S_5\) includes the first five terms: \(1 - \frac{x}{2} + \frac{x^2}{4} - \frac{x^3}{8} + \frac{x^4}{16}\).

Calculating these gives students a micro view of how a portion of the series behaves, so they can anchor their understanding in smaller sections before examining the infinite series. This approach is crucial in analyzing convergence and testing the behavior of infinite series.

Graphing Utility

Graphing utilities, such as graphing calculators or software, are invaluable tools for visualizing complex functions and their behavior. When graphing the sum function \(\frac{2}{2+x}\), alongside the partial sums \(S_3\) and \(S_5\), it becomes clear how quickly the series converges to the function.
By plotting:

• The function \(\frac{2}{2+x}\), which represents the sum of the infinite series.
• The partial sums \(S_3\) and \(S_5\) on the same graph.

Students can observe how the partial sums approach the curve of the function with increasing terms. This graphical representation offers a clear visual understanding that numerical methods or algebraic symbolism might not fully convey.

Sum of Infinite Series

The sum of an infinite series provides a way of evaluating the entire series as it extends indefinitely. In geometric series, this is possible only when the absolute value of the common ratio \(r\) is less than 1, which ensures convergence.
For this series, using the identified common ratio \(-\frac{x}{2}\), we express the sum as:\[ S = \frac{a}{1 - r} = \frac{1}{1 + \frac{x}{2}} = \frac{2}{2 + x} \]This expression gives the complete value the series converges to as more terms are added. Understanding this concept helps to comprehend that not all infinite sums equal infinity. Instead, they can have finite values. This understanding not only applies to theoretical problems but also to practical applications across physics, economics, and beyond, where series are prevalent.