Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-24, use the methods of Example 5 to find power series in \(x\) for each function \(f\). $$ f(x)=\left(\tan ^{-1} x\right)\left(1+x^{2}+x^{4}\right) $$

Step-by-Step Solution

Answer

The power series for \( f(x) = (\tan^{-1}x)(1 + x^2 + x^4) \) is \( x + \frac{2x^3}{3} + \frac{17x^5}{15} + \cdots \).

Step 1: Identify the series for different functions

We will first find the power series representation for each component function: \( \tan^{-1}(x) \) and \( 1 + x^2 + x^4 \) individually.

Step 2: Power Series for \( \tan^{-1}(x) \)

The power series expansion for \( \tan^{-1}(x) \) about 0 is derived from its Taylor series. The series is \( \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \). This series converges for \( |x| \leq 1 \).

Step 3: Power Series for \( 1 + x^2 + x^4 \)

The function \( 1 + x^2 + x^4 \) is already in a series form, where each term is simply expressed as a polynomial. Its power series is simply its polynomial representation.

Step 4: Multiply the series

Now, multiply the power series for \( \tan^{-1}(x) \) by \( 1 + x^2 + x^4 \). Use the distributive property to multiply each term in the first series by each term in the second series. This will give \( \left(x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots \right)\left(1 + x^2 + x^4\right) \).

Step 5: Compute the product series

To compute the product, distribute each term from \( \tan^{-1}(x) \) through \( 1 + x^2 + x^4 \):- \( x(1 + x^2 + x^4) = x + x^3 + x^5 \)- \( -\frac{x^3}{3}(1 + x^2 + x^4) = -\frac{x^3}{3} - \frac{x^5}{3} - \frac{x^7}{3} \)- \( \frac{x^5}{5}(1 + x^2 + x^4) = \frac{x^5}{5} + \frac{x^7}{5} + \frac{x^9}{5} \)Add these series together.

Step 6: Combine like terms

Combine like terms from the multiplication step to find the power series. - Terms with \( x: x \)- Terms with \( x^3: x^3 - \frac{x^3}{3} = \frac{2x^3}{3} \)- Terms with \( x^5: x^5 - \frac{x^5}{3} + \frac{x^5}{5} = \frac{17x^5}{15} \)- Higher order terms can be calculated similarly.

Key Concepts

Power Series

A power series is an infinite series of the form \( a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots \). It can be thought of as an infinite polynomial and is a powerful tool in calculus. Power series have a radius of convergence within which they accurately represent a function.
They are used to approximate functions, calculate limits, and solve differential equations.
For a given function, the coefficients \( a_n \) determine the contribution of each term \( x^n \) to the sum of the series. This is especially useful when working with functions that are difficult to compute directly.

• Each component function in a given problem can have its power series.
• In the exercise, both components \( \tan^{-1}(x) \) and \( 1 + x^2 + x^4 \) were expanded into their power series form.

Understanding power series is essential in recognizing how seemingly complex functions can be broken down into simpler polynomial-like components.

Taylor Series

The Taylor series is a special type of power series that represents functions as an infinite sum of derivative terms at a single point. For a function \( f(x) \), its Taylor series around a point \( a \) is given by:
\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \]
This series makes it possible to approximate functions using polynomials based on their derivatives.

• The Taylor series allows for approximations over a broader range than a simple power series when it converges.
• In the given problem, the series for \( \tan^{-1}(x) \) is obtained via its Taylor series around 0.

Such approximations are useful in practical calculations where exact solutions are tough to find or unnecessary.

Polynomial

A polynomial is a mathematical expression involving terms with variables raised to whole-number exponents. Polynomials take the form:
\( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \)
Each term in a polynomial is called a monomial, and the highest power of \( x \) determines the polynomial's degree. Polynomials are fundamental in algebra and appear in many areas of mathematics.
In calculus, polynomials often approximate more complex functions, especially near a specific point, making them integral to solving calculus problems.

• In the provided exercise, the function \( 1 + x^2 + x^4 \) is a straightforward example of a polynomial.
• It is used as the multiplier in the power series, showing the simplicity of polynomials when combined with series expansions.

Understanding polynomials is crucial because they serve as the building blocks for more intricate mathematical expressions depicted by series.

Arctangent Function

The arctangent function, denoted as \( \tan^{-1}(x) \), is the inverse of the tangent function. It yields the angle whose tangent is \( x \). For real numbers, its range is \(-\pi/2\) to \(\pi/2\), a range important in trigonometry and calculus.
The power series representation of \( \tan^{-1}(x) \) is derived from its Taylor series, which provides a polynomial-like approximation over \(|x| \leq 1\). This function is particularly meaningful in integrating, solving equations, and approximating values.

• For example, the series: \( \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots \), simplifies calculations by offering an infinite set of terms representing the arctangent.
• In the problem, it plays a key role in forming the final power series through the multiplication of its series by the polynomial \( 1 + x^2 + x^4 \).

Mastering the arctangent function and its series expands both practical and theoretical calculus skills, valuable in scientific computations.