Problem 14 - Essential Calculus Early Transcendentals

Question

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=1 / x, \quad a=-3$$

Step-by-Step Solution

Answer

The Taylor series is \(-\frac{1}{3} - \frac{1}{9}(x+3) - \frac{1}{54}(x+3)^2 - \frac{1}{486}(x+3)^3 + \ldots\).

Step 1: Understand the Taylor series formula

The Taylor series for a function \( f(x) \) centered at \( a \) is given by: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \] This series represents the function as an infinite sum of its derivatives evaluated at \( a \) and multiplied by powers of \( (x-a) \).

Step 2: Calculate the function value at \(a\)

Evaluate \( f(x) = \frac{1}{x} \) at \( a = -3 \). \[ f(-3) = \frac{1}{-3} = -\frac{1}{3} \]

Step 3: Calculate the first derivative and evaluate at \(a\)

Find the first derivative: \[ f'(x) = -\frac{1}{x^2} \] Evaluate it at \( a = -3 \): \[ f'(-3) = -\frac{1}{(-3)^2} = -\frac{1}{9} \]

Step 4: Calculate the second derivative and evaluate at \(a\)

Find the second derivative: \[ f''(x) = \frac{2}{x^3} \] Evaluate it at \( a = -3 \): \[ f''(-3) = \frac{2}{(-3)^3} = -\frac{2}{27} \]

Step 5: Calculate the third derivative and evaluate at \(a\)

Find the third derivative: \[ f'''(x) = -\frac{6}{x^4} \] Evaluate it at \( a = -3 \): \[ f'''(-3) = -\frac{6}{(-3)^4} = -\frac{2}{27} \]

Step 6: Write out the first few terms of the Taylor Series

Use the calculated values to write the Taylor series.\[ f(x) = -\frac{1}{3} - \frac{1}{9}(x+3) - \frac{1}{54}(x+3)^2 - \frac{1}{486}(x+3)^3 + \ldots \]Each term is calculated using the evaluated derivatives and powers of \( (x - (-3)) = (x+3) \).

Step 7: Construct the general term of the series

Observe the pattern in derivatives and construct the general term. This involves finding the general expression for the \( n^{th} \) derivative and using it in the series formula. The general term is: \[ f^{(n)}(-3) = (-1)^{n+1} \frac{n!}{(-3)^{n+1}} \] So, the series is: \[ \sum_{n=0}^{\infty} (-1)^{n+1}\frac{n!}{(-3)^{n+1}} (x+3)^n \]

Key Concepts

Power Series Expansion

In mathematics, a power series expansion is a way of representing a function as an infinite sum of terms. These terms are expressed in the form of powers of a variable. A Taylor series is a specific type of power series that represents a function around a particular point, called the center. In our example, the function given is \(f(x) = \frac{1}{x}\) with a center at \(a = -3\).

A Taylor series takes the general form:

• \(f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \)

The power series expansion involves calculating successive derivatives of the function, evaluating those derivatives at the center, and constructing terms using these values. Each term includes a derivative, a factorial, and the variable \((x-a)\) raised to increasing powers.

These expansions can approximate the behavior of functions over intervals and are also used in solving complex mathematical problems.

Derivatives

Derivatives measure how a function changes as its input changes. In the context of Taylor series, derivatives are used to capture the various degrees of slopes or rates of changes of the function around a particular point.

For the function \( f(x) = \frac{1}{x} \), the derivatives are:

• First derivative: \( f'(x) = -\frac{1}{x^2} \)
• Second derivative: \( f''(x) = \frac{2}{x^3} \)
• Third derivative: \( f'''(x) = -\frac{6}{x^4} \)

Each of these derivatives is evaluated at \( a = -3 \), which simplifies the expressions for terms in the Taylor series.

Derivatives provide the necessary information to determine the coefficients of the series and predict the behavior of functions around the chosen center.

Infinite Sum

An infinite sum, or series, is a sum that continues indefinitely. In the case of the Taylor series, the function \( f(x) \) is expressed as an infinite sum of its terms, which involve the derivatives evaluated at the center, factorials, and powers of \( (x - a) \).

The general term for a Taylor series can be written as:

• \( f^{(n)}(a) \cdot \frac{(x-a)^n}{n!} \)

Here, this contributes to forming:

• \( \sum_{n=0}^{\infty} f^{(n)}(a) \cdot \frac{(x-a)^n}{n!} \)

The infinite nature of this series allows the function to be accurately represented, or approximated, over a range of its input. Even though theoretically infinite, in practical applications, Taylor series are often truncated after a few terms to make them manageable while still sufficiently accurate.

Understanding infinite sums is fundamental for utilizing Taylor series effectively to solve real-world problems and perform complex mathematical analyses.