Problem 3 - Calculus Early Transcendentals

Question

How do you find the coefficients of the Taylor series for \(f\) centered at \(a ?\)

Step-by-Step Solution

Answer

Answer: To find the coefficients of the Taylor series, follow these steps: 1. Understand the concept of a Taylor series. 2. Compute the n-th derivative of function f. 3. Evaluate the n-th derivative at point a. 4. Calculate the coefficients using the formula: \(c_n = \frac{f^{(n)}(a)}{n!}\). 5. Write down the Taylor series using the general formula and calculated coefficients.

Step 1: Understand what a Taylor series is

A Taylor series is a representation of a function as an infinite sum of terms calculated based on the function's derivatives at a single point. The general formula for the Taylor series is given by: $$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$ Where \(f^{(n)}(a)\) denotes the n-th derivative of the function evaluated at point \(a\), and \(n!\) represents the factorial of n.

Step 2: Take the n-th derivative of the function f

In order to find the Taylor series coefficients, we need to compute the derivatives of the function \(f\) at the center point \(a\) up to the desired order. If we want the general coefficients, we will need the n-th derivative of the function, represented as \(f^{(n)}(x)\).

Step 3: Evaluate the n-th derivative at the point a

Now we need to find the value of the n-th derivative at the point \(a\). This is done by simply substituting \(x = a\) in the expression for the n-th derivative: \(f^{(n)}(a)\).

Step 4: Calculate the Taylor series coefficients

The coefficients of the Taylor series can now be computed using the formula: $$c_n = \frac{f^{(n)}(a)}{n!}$$ For each term in the Taylor series, the coefficient \(c_n\) is obtained by evaluating the \(f^{(n)}(a)\) value and then dividing it by n!.

Step 5: Write down the Taylor series

Finally, we write down the Taylor series for the given function f using the general formula and the calculated coefficients: $$f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$

Key Concepts

Derivatives

Derivatives play a crucial role in understanding Taylor series. In essence, a derivative measures how a function changes as its input changes. When finding Taylor series, we use derivatives to approximate the behavior of a function around a specific point, known as the center.

• The first derivative (\(f'(a)\)) gives us the slope of the function at point \(a\).
• Higher-order derivatives, like the second derivative (\(f''(a)\)) and third derivative (\(f'''(a)\)), provide more detailed information about the curvature of the function.

Computing these derivatives accurately allows us to build a series of polynomial terms that collectively approximate the overall function. This is the core of the Taylor series' utility: using derivatives to construct an approximation of a desired function.

Factorial

A factorial, denoted by the symbol (\(!\)), is a mathematical operation used extensively in the context of Taylor series. The factorial of a number \(n\) is the product of all positive integers from \(1\) to \(n\).

• For instance, \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• The factorial is used to normalize the size of the terms in a Taylor series, ensuring each term's contribution reflects the degree of its derivative.

In the formula for Taylor series, the factorial appears as the denominator in the terms \(\frac{f^{(n)}(a)}{n!}(x-a)^n\), helping to scale down the impact of each successive term. This scaling is crucial for convergence and overall accuracy of the approximation.

Infinite Series

An infinite series is a sum of infinitely many terms. In the Taylor series context, it refers to the way we represent a function as an endless sum. Although it sounds daunting, the series converges to the function over an interval when properly managed.

• The infinite series involves terms like \(\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\), combining derivatives, factorials, and powers.
• When summing these terms, the result approximates the function, often converging closely except at certain special points.

Understanding that the series goes on indefinitely, but still converges, is pivotal. It highlights the range within which the Taylor series effectively serves as a perfect approximation of the function.

Mathematical Coefficients

Mathematical coefficients in Taylor series are the multipliers of each term in the series expansion. These coefficients are derived from the function’s derivatives, as seen in the formula for Taylor series, where each coefficient is \(c_n = \frac{f^{(n)}(a)}{n!}\).

• The coefficient \(c_0\) is simply the value of the function at \(a\), \(f(a)\).
• Subsequent coefficients depend progressively on higher order derivatives and their respective factorials.

These coefficients determine the significance of each polynomial term within the series, assisting in how closely the series matches the function. Proper calculation of the coefficients ensures each term correctly reflects the influence of its corresponding derivative and polynomial degree, leading to an accurate representation of the original function.