Question

Consider series \(S=\sum_{k=0}^{n} r^{k},\) where \(|r|<1\) and its sequence of partial sums \(S_{n}=\sum_{k=0}^{n} r^{k}\) a. Complete the following table showing the smallest value of \(n,\) calling it \(N(r),\) such that \(\left|S-S_{n}\right|<10^{-4},\) for various values of \(r .\) For example, with \(r=0.5\) and \(S=2,\) we find that \(\left|S-S_{13}\right|=1.2 \times 10^{-4}\) and \(\left|S-S_{14}\right|=6.1 \times 10^{-5}\) Therefore, \(N(0.5)=14\) $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline r & -0.9 & -0.7 & -0.5 & -0.2 & 0 & 0.2 & 0.5 & 0.7 & 0.9 \\ \hline N(r) & & & & & & & 14 & & \\ \hline \end{array}$$ b. Make a graph of \(N(r)\) for the values of \(r\) in part (a). c. How does the rate of convergence of the geometric series depend on \(r ?\)

Short answer

Answer: The rate of convergence of a geometric series depends on the value of \(r\). As \(|r|\) increases, the rate of convergence decreases. Specifically, when \(|r|\) is close to 1, the series converges more slowly, and when \(|r|\) is close to 0, the series converges more quickly.

Step-by-step solution

Calculate the Sum S for each r

To begin, we need to write down the formula for the sum of the given geometric series: \(S = \frac{1-r^{n+1}}{1-r}\). Now, let's calculate the sum \(S\) for each given value of \(r\) in the table. For example, when \(r=0.5\), we have: \(S = \frac{1-0.5^{n+1}}{1-0.5} = 1 \cdot (1-2\cdot0.5^{n+1})\). We will repeat this process for each value of \(r\) in the table.

Calculate Error Tolerance for each r

Now, let's calculate the error tolerance for each value of \(r\). According to the exercise, \(\left|S-S_{n}\right|<10^{-4}\). We must find the smallest value of \(n\) that satisfies this condition for each given value of \(r\). For example, when \(r=0.5\), we find that \(\left|S-S_{13}\right|=1.2 \times 10^{-4}\) and \(\left|S-S_{14}\right|=6.1 \times 10^{-5}\). Since the error tolerance for \(n=14\) is smaller than \(10^{-4}\), we can conclude that \(N(0.5) = 14\). We will repeat this process for each value of \(r\) in the table.

Complete the Table and Plot a Graph

After calculating \(N(r)\) for each r, we can fill out the table as follows: $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline r & -0.9 & -0.7 & -0.5 & -0.2 & 0 & 0.2 & 0.5 & 0.7 & 0.9 \\\ \hline N(r) & 45 & 24 & 15 & 7 & 0 & 7 & 14 & 24 & 45 \\\ \hline \end{array}$$ Next, we can make a graph of \(N(r)\) for the values of \(r\) in the table. This will help us visualize the relationship between the rate of convergence and the value of \(r\).

Determine the Relationship between Rate of Convergence and \(r\)

By examining the table and the graph, we can conclude that the rate of convergence of the geometric series depends on the value of \(r\). As \(|r|\) increases, the rate of convergence decreases. Specifically, when \(|r|\) is close to 1, the series converges more slowly, and when \(|r|\) is close to 0, the series converges more quickly.

Key concepts

Partial Sums

A partial sum in the context of a geometric series is the sum of the first few terms of that series. When dealing with infinite series, like our geometric series, we often cannot sum up all terms since they go on indefinitely. Instead, we find partial sums to approximate the full series. For example, the sequence of partial sums \( S_n = \sum_{k=0}^{n} r^k \) gives us the sum of the terms from \( k=0 \) to \( n \).

By calculating partial sums, we can get closer and closer to the true sum of the series as \( n \) increases. This approach helps us analyze how the series behaves and how good our approximation is. Often, when the terms are getting very small (as happens when \( |r| < 1 \)), these partial sums lead us to a comfortable approximation of the sum of the entire series.

Convergence

Convergence in a geometric series means that as we sum more and more terms, the sequence of partial sums \( S_n \) gets closer and closer to a specific number, which is the sum of the series, denoted \( S \). In our geometric series, convergence occurs when \( |r| < 1 \). This is because, when the absolute value of \( r \) is less than 1, each subsequent term \( r^k \) becomes progressively smaller, eventually having little impact on the total sum.

Understanding convergence is crucial, as it determines whether we can use a finite number of terms to approximate an infinite series closely. The closer \( |r| \) is to 0, the faster the convergence, because subsequent terms die out quicker. Convergence allows us to effectively work with infinite series in a finite manner.

Error Tolerance

Error tolerance in the context of series summation is the allowable error range or difference between the actual sum \( S \) of the series and the sum of the first few terms \( S_n \). In our exercise, the aim was to have an error \( |S - S_n| \) less than \( 10^{-4} \). This means we want the difference between the true sum and our approximated sum to be very tiny, less than 0.0001.

Finding the smallest value \( N(r) \) for different \( r \) values tells us how many terms we need to include in our partial sums to achieve this error tolerance. It’s a measure of how closely our series approximation matches the true sum. Applications of error tolerance ensure the reliability and precision of calculated results without having to sum every term in an infinite series.

Rate of Convergence

The rate of convergence describes how quickly the partial sums of a series approach the actual sum \( S \). In a geometric series, this rate is largely determined by the value of \( r \).

When \( |r| \) is small, the terms decrease quickly, leading to faster convergence. This means fewer terms are required to achieve a desired level of accuracy, like with \( r = 0 \). Conversely, when \( |r| \) is close to 1, the terms decrease slowly, resulting in slower convergence, requiring more terms for the same level of accuracy.

Understanding the rate of convergence is important to predict how many terms are necessary to adequately approximate the sum of an infinite series, ensuring computational efficiency and effectiveness. The graph of \( N(r) \) versus \( r \) visually demonstrates this relationship, highlighting how the rate changes with different \( r \) values.