Question

Determine the Taylor series about the point \(x_{0}\) for the given function. Also determine the radius of convergence of the series. \(x^{2}, \quad x_{0}=-1\)

Short answer

Based on the given step-by-step solution, the Taylor series of the function \(f(x) = x^2\) about the point \(x_{0} = -1\) is \(2x^2 + 2x\), and the radius of convergence is infinite.

Step-by-step solution

Find the n-th derivative of the function

For the given function \(f(x)=x^2\), we can start by finding the first few derivatives of the function with respect to x: \(f'(x) = 2x\) \(f''(x) = 2\) \(f'''(x) = 0\) We can easily observe that all the higher order derivatives beyond the second one will be 0.

Determine the Taylor series about the point \(x_{0} = -1\)

To find the Taylor series, we can use the formula for the n-th term of the Taylor series centered around \(x_0\): \(a_n = \dfrac{f^{(n)}(x_0)}{n!} (x-x_0)^n\) We already found the n-th derivatives of \(f(x)\) in step 1. Now, let's plug them into the formula: \(a_0 = \dfrac{f(-1)}{0!} (x-(-1))^0 = 1\) \(a_1 = \dfrac{f'(-1)}{1!} (x-(-1))^1 = -2(x+1)\) \(a_2 = \dfrac{f''(-1)}{2!} (x-(-1))^2 = 2(x+1)^2\) Since after n=2 all the higher derivatives are zero, there won't be any higher terms in the Taylor series. So, the Taylor series for \(f(x)=x^2\) around \(x_0=-1\) is: \(1 - 2(x+1) + 2(x+1)^2 = 1 - 2x - 2 + 2x^2 + 4x + 2 = 2x^2+2x\).

Find the radius of convergence

For this function, the Taylor series is a polynomial, meaning that there won't be an infinite number of terms. Therefore, the radius of convergence for this series will be infinite. The function converges for all x in the real number line. To summarize, the Taylor series of \(f(x)=x^2\) about the point \(x_{0}=-1\) is \(2x^2+2x\), and the radius of convergence is infinite.

Key concepts

Radius of Convergence

Understanding the radius of convergence is essential when working with power series, as it defines the set of values for which the series converges. For any power series \(\sum a_n(x-x_0)^n\), the radius of convergence (R) is the distance from the center \(x_0\) to the nearest point where the series ceases to converge. A series can converge at points within this radius, while it may diverge if attempting to go beyond.

To determine the radius of convergence, various tests such as the Ratio Test, Root Test, or comparison with known series can be used. In our exercise, the Taylor series for the function \(f(x)=x^2\) about the point \(x_0=-1\) is \(2x^2+2x\), which is a polynomial with a finite number of terms. Since a polynomial does not have terms continuing indefinitely, the radius of convergence is infinite, indicating that the series converges for all real numbers.

Nth Derivative

The process of differentiation can be extended beyond the first or second derivative to the nth derivative. The nth derivative of a function represents the rate of change of the \( (n-1) \)-th derivative with respect to the independent variable, often denoted as \( f^{(n)}(x) \).

For example, if starting with the function \( f(x) = x^2 \), the first derivative is \( f'(x) = 2x \), and the second derivative is \( f''(x) = 2 \). As we continue this process, we observe that all successive derivatives after \( f''(x) \) will be zero. This property of derivatives ceasing to produce new non-zero terms is crucial when forming Taylor series, as it determines the number of non-zero terms in the expansion.

Power Series

A power series is an infinite series of the form \(\sum a_n(x-x_0)^n\), where \(a_n\) represents the series coefficients, \(x\) is the variable, and \(x_0\) is the center of the series. Power series are highly valued for their role in approximating functions through an infinite sum of polynomial terms. Each term in a power series can be viewed as a contribution to the function's approximation near the center \(x_0\).

Power series demonstrate a wide array of behaviors when it comes to convergence. Depending on the value of \(x\), a power series may converge to a finite number, diverge, or oscillate. In the context of our exercise, the Taylor series formed for the function \( f(x)=x^2 \) at \( x_0=-1 \) is also a power series that just happens to terminate, simplifying its convergence analysis—as it's a polynomial, convergence is guaranteed on the entire real line.

Convergence of Series

The convergence of a series refers to whether the sequence of partial sums of the series approaches a finite limit as its number of terms goes to infinity. Identifying whether a series converges is critical in multiple areas of mathematics and physics, as it assures that sums of infinitely many terms make sense and can be used in calculations.

To determine the convergence of a series, various tests are employed, such as the Integral Test, Comparison Test, Ratio Test, or Alternating Series Test. In our exercise's case, since the Taylor series produces a polynomial, the question of convergence is straightforward; with finite terms, the series will converge to the function itself within the infinite radius of convergence. In other scenarios where series are not finite, careful testing is needed to assess their behavior.