Question

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(\left|R_{n}\right|<10^{-6}\) ). Functions defined as series Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty}(-1)^{k} x^{k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Short answer

Question: Determine the number of terms needed to ensure the partial sum of the series \(f(x)=\sum_{k=0}^{\infty}(-1)^{k} x^{k}\) satisfies \(\left|R_{n}\right|<10^{-6}\), evaluate the function at \(x = 0, 0.2, 0.5\), and determine the domain of the function. Short Answer: To ensure the partial sum of the series satisfies \(\left|R_{n}\right|<10^{-6}\), for each given value of \(x\), find the smallest integer \(n\) that satisfies the inequality \(\left|\frac{(-x)^{n}}{1 - (-x)}\right|<10^{-6}\). Evaluating the function for the given values of \(x\), we have \(f(0) = 0\), and for \(x=0.2\) and \(x=0.5\), use the determined number of terms to compute the partial sums using the geometric series formula. The domain of the function \(f(x)\) is \(-1<x<1\).

Step-by-step solution

1. Determine the general term for the geometric series.

The general term for a geometric series is given by the formula \(a_n = a_1r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. In this case, we have \(a_1 = 1\) and the common ratio \(r = -x\). Therefore, the general term for this geometric series is \(a_n = (-x)^{n-1}\).

2. Determine the error term formula for the series.

The error term formula for a convergent geometric series is given by \(R_n = \frac{a_{n+1}}{1 - r}\), where \(R_n\) is the error term and we have to find the least integer \(n\) for which \(\left|R_{n}\right|<10^{-6}\). We will replace \(a_{n+1}\) and \(r\) with the values we found in step 1. Therefore, the error term formula for this series is \(R_n = \frac{(-x)^{n}}{1 - (-x)}\).

3. Use the error term to find the number of terms needed.

We want to find the least integer \(n\) such that \(\left|R_{n}\right|<10^{-6}\). Using the formula for \(R_n\) from step 2, we have \(\left|\frac{(-x)^{n}}{1 - (-x)}\right|<10^{-6}\). To solve this inequality, we simply need to find the least integer \(n\) that satisfies it for each given value of \(x\).

4. Evaluate the function at the given points.

Now, we can evaluate the function \(f(x)\) at the given values using the number of terms found in step 3: - For \(f(0)\), the series evaluates to 0. - For \(f(0.2)\), find the smallest integer \(n\) that satisfies the inequality from step 3 and evaluate the partial sum using the geometric series formula. - For \(f(0.5)\), repeat the process to find the corresponding partial sum. - For \(f(1)\) and \(f(1.5)\), the series does not converge, which we will explain in the next step.

5. Determine the domain of the function \(f(x)\).

The geometric series converges when \(|r|<1\). In our case, the common ratio \(r=-x\). Therefore, the series converges when \(|-x|<1\), which implies that the domain of the function \(f(x)\) is \(-1<x<1\).

Key concepts

Partial Sum

In the context of a geometric series, the partial sum is the sum of the first few terms, certain up to a specified term, of the series. It helps to approximate the total value of a series by calculating a limited number of its terms. For the series in this exercise, the partial sum is represented by adding terms until the sum is close enough to the actual value of the series. The goal is to keep this partial sum within a given error margin. In this exercise, the error margin is less than \(10^{-6}\).

Finding how many terms to include in the partial sum requires us to satisfy the inequality \(|R_n| < 10^{-6}\). This determines the least number of terms \(n\) required for each value of \(x\), ensuring the partial sum is accurate to within \(10^{-6}\) of the series' value.

• The error term \(R_n\), in this case, ensures the proximity of the partial sum to the actual value of the series.
• When working with geometric series, solving for the partial sum allows practitioners to determine efficiency in calculating complex series to an accepted degree of accuracy.

Convergence

Convergence in series is when the infinite sum of terms approaches a finite limit. In our context, a geometric series converges provided that the absolute value of the common ratio \(|r|\) is less than 1. When dealing with convergence:

For the function defined by the series \(f(x) = \sum_{k=0}^{\infty}(-1)^{k} x^{k}\), it only converges when \(|-x| < 1\). Therefore, when \(x\) values like 1 and 1.5 are put into the function, the series diverges because \(|r|\) is not less than 1. This means the series would not reach a finite limit, making it impossible to compute a meaningful sum.

• Convergence is vital because it determines if the series yields a meaningful result.
• Understanding convergence ensures the correct representation of the function within its applicable range.

Recognizing convergence conditions helps us practically in deciding for which \(x\)-values the function \(f(x)\) is valid.

Domain of a Function

The domain of a function represents all possible input values (or \(x\)-values) for which the function is defined. Here, we find the domain by determining where the series \( f(x) = \sum_{k=0}^{\infty}(-1)^{k} x^{k} \) converges.

Since the series uses the common ratio \(r = -x\), convergence requires \(|-x| < 1\). This simplifies to \(-1 < x < 1\). Thus, the geometric series, and consequently the function \(f(x)\), is defined only for \(x\)-values within this interval. Any \(x\) outside this range results in divergence, making the function undefined.

• The domain reflects the boundary within which the series retains meaning and utility.
• Conceptually, it provides the range of values for which \(f(x)\) can give accurate results.

Knowing the domain is crucial in avoiding errors and ensuring the function is used correctly for its intended purpose.