Problem 77 - Calculus Early Transcendentals

Question

Consider the following function and its power series: $$ f(x)=\frac{1}{(1-x)^{2}}=\sum_{k=1}^{\infty} k x^{k-1}, \quad \text { for }-1

Step-by-Step Solution

Answer

Based on the given step-by-step solution, we can create the following short answer: First, plot the original function (\(f(x)\)) and its approximations (\(S_5(x)\) and \(S_{10}(x)\)) using graphing software like Desmos, and compare their behavior at sample points (\(x = -0.9, -0.8, \ldots, -0.1,0,0.1, \ldots, 0.8,0.9\)). Next, to find the value of \(n\) that guarantees an error less than 0.01 at all the sample points, apply the Lagrange remainder formula and solve for \(n\). This would require computing the derivatives of the function up to the \((n+1)\)-th derivative. Finally, find the maximum of the absolute value of the derivatives, substitute them in the Lagrange remainder formula, and solve for \(n\) to guarantee the error \(\left|f(x)-S_{n}(x)\right|<0.01\) at all the sample points.

Step 1: Plot the Functions

We are given the function \(f(x)=\frac{1}{(1-x)^{2}}\) and its power series representation. We need to plot \(f(x)\) and its approximations at the sample points \(x = -0.9, -0.8, \ldots, -0.1,0,0.1, \ldots, 0.8,0.9\). We are interested in the behavior of the approximations for \(n=5\) and \(n=10\). To plot the functions, use graphing software like Desmos, for example.

Step 2: Compare the Graphs

Study both graphs, and observe where the difference between \(f(x)\) and \(S_n(x)\) is the greatest. You may use built-in graphing software tools to measure the differences. ##Part b##

Step 3: Upper Bound for the Remainder Term

Using the remainder term, let's find the value of \(n\) required for \(\left|f(x)-S_{n}(x)\right|<0.01\) at all sample points. Since the remainder term for a Taylor series represents the error, we need to ensure that the error converges to zero with increasing value of \(n\). In order to guarantee this, we can use the formula for the error term of a power series (Lagrange remainder formula). Note that in this case, the function \(f(x)\) is at least \(n+1\) times continuously differentiable, giving us the ability to use this formula.

Step 4: Lagrange Remainder Formulation

To determine \(n\), apply the Lagrange remainder formula and solve for it: $$ \left|f(x) - S_n(x)\right| = \left|\frac{f^{(n + 1)}(c)}{(n + 1)!}x^{n + 1}\right| < 0.01 $$ Where \(c\) is a value between the sample points.

Step 5: Differentiate the Function

Compute the derivatives of the function up to the \((n+1)\)-th derivative to find the maximum of the absolute value of the derivatives within the range of \(x\).

Step 6: Find the Value of \(n\) Required

Once you find the maximum of the absolute value of the derivatives, substitute them in the Lagrange remainder formula, and solve for \(n\) to guarantee the error \(\left|f(x)-S_{n}(x)\right|<0.01\) at all the sample points.

Key Concepts

Taylor Series

The Taylor Series is a powerful mathematical tool used to represent functions as an infinite sum of terms calculated from the values of their derivatives at a single point. In a Taylor Series, the function is expressed as a power series, which means it converges to the function within a certain interval whenever certain conditions are met. For the function given in our exercise, the Taylor Series expansion is represented as: \( f(x)=\frac{1}{(1-x)^{2}}=\sum_{k=1}^{\infty} k x^{k-1} \), valid for \(-1 < x < 1\). - **Power Series Representation:** A power series in the form of \(a_0 + a_1x + a_2x^2 + \,\ldots\) can effectively approximate functions within a specific interval.- **Function Approximation:** By summing a finite number of terms, \(S_n(x)\), we can create polynomial approximations that represent \(f(x)\) closely within the specified range.In our exercise, we observe that the length of the Taylor series (denoted by \(n\)) affects how closely \(S_n(x)\) can approximate the original function \(f(x)\). As \(n\) increases, \(S_n(x)\) becomes a better approximation of \(f(x)\). Understanding how Taylor Series function helps generate accurate polynomial approximations is key.

Lagrange Remainder Formula

The Lagrange Remainder Formula provides a bound for the error or remainder of a Taylor Series approximation. This formula is crucial when determining how many terms \(n\) are needed for a satisfactory approximation.To understand how this applies to our task, note that the remainder of a Taylor series, \(R_n(x)\), helps us determine the error between the actual function \(f(x)\) and the approximation \(S_n(x)\). It is given by: \[ R_n(x) = \frac{f^{(n + 1)}(c)}{(n + 1)!} x^{n + 1} \]where \(f^{(n+1)}(c)\) is the \((n+1)\)-th derivative of the function evaluated at some point \(c\) within the interval (-1, 1).- **Error Estimation:** The formula allows us to estimate the maximum error at any given point. Knowing the error helps us decide the number of terms \(n\) necessary to meet a given tolerance level, for example, \(\left|f(x)-S_n(x)\right|<0.01\).- **Practical Usage:** This method ensures our approximation meets specified accuracy by appropriately adjusting \(n\).

Graphing Functions

Graphing functions play a crucial role in visualizing how well the polynomial approximation \(S_n(x)\) represents the original function \(f(x)\). By graphing both \(f(x)\) and \(S_n(x)\), we can easily identify areas where the approximation deviates significantly from the actual function.To compare the behavior of \(f(x)\) and its approximation \(S_n(x)\) for different values of \(n\) (e.g., \(n=5\) and \(n=10\)), graphing software such as Desmos can be used.- **Comparison:** Graph both \(f(x)\) and \(S_n(x)\) at specified sample points \(x = -0.9, -0.8, \ldots, -0.1, 0, 0.1, \ldots, 0.8, 0.9\).- **Identifying Deviations:** Visual inspection of the graphs allows you to see where the error is greatest—typically near the series boundaries, which is crucial for understanding the efficacy of your approximation and where improvement is needed.By assessing these graphs, you gain insights into how many terms (\(n\)) might be necessary for an accurate approximation throughout the entire interval.

Polynomial Approximation

Polynomial approximation is the idea of using a polynomial to estimate a function's value within a specific range or interval. This concept is pivotal when dealing with Taylor Series, as it directly involves approximating a complicated function with a simpler polynomial.For the function \(f(x)\) in this exercise, we use the finite sum of the Taylor Series as a polynomial approximation, \(S_n(x)\). The process of creating such approximations involves analyzing:- **Choice of Degree (\(n\)):** This determines how many terms are used, affecting the accuracy of the approximation.- **Convergence:** The polynomial \(S_n(x)\) must converge towards \(f(x)\) within the desired interval.The challenge in polynomial approximation is achieving a balance between computational feasibility and the desired level of accuracy. More terms in the polynomial can result in higher accuracy, but also more complexity. In the context of the exercise, the focus lies in finding the right number \(n\) such that the difference \(|f(x) - S_n(x)| < 0.01\) holds true across all given sample points. Thus, understanding polynomial approximation involves grasping the importance of balancing term count and error threshold.