Problem 10 - Calculus: Early Transcendentals

Question

Find a power series representation for the function and determine the interval of convergence. \( f(x) = {x + a}{x^2 + a^2}, a > 0 \)

Step-by-Step Solution

Answer

The power series is \( x^3 + ax^2 + a^2x + a^3 \) with convergence \((-\infty, \infty)\).

Step 1: Expand the Expression

First, expand the expression given by multiplying out the terms: \( f(x) = (x+a)(x^2 + a^2) = x^3 + ax^2 + a^2x + a^3 \). This is the polynomial form of the function.

Step 2: Consider Series Representation

We need to express \( f(x) \) as a power series centered around zero. Since \( f(x) \) is already a polynomial of degree three, its power series representation will be the same as its polynomial form: \( f(x) = x^3 + ax^2 + a^2x + a^3 \).

Step 3: Determine Convergence Interval

Since the power series representation of \( f(x) \) is a finite polynomial rather than an infinite series, it converges for all \( x \). Therefore, the interval of convergence is \( (-\infty, \infty) \).

Key Concepts

Interval of Convergence

In calculus, determining the interval of convergence for a power series is crucial to understanding where the series accurately represents a function. The interval of convergence is the set of all values of

• \(x\) for which the series converges, producing a valid representation of the function.
• For our problem, where the function \(f(x) = (x + a)(x^2 + a^2)\) is expanded to a polynomial form \(x^3 + ax^2 + a^2x + a^3\), we find that it's a finite polynomial.

Finite polynomials are special because they automatically converge for all real numbers. Unlike infinite series that may converge only on a limited interval or specific set of points, finite polynomials do not have restrictions in this sense. Therefore, no matter the value of \(x\), the polynomial will always yield a valid output. Hence, for this polynomial, the interval of convergence is

• \((-\infty, \infty)\), meaning it covers every real number.

Knowing about intervals of convergence helps in understanding the domain and limitations of functions and series.

Polynomial Expansion

Polynomial expansion involves rewriting expressions in a simpler form. Using basic algebraic principles, we multiply terms to remove parentheses and express the function as a sum of terms with coefficients and powers of the variable. In our case, we begin with the expression \((x + a)(x^2 + a^2)\), and expand it to obtain

• \(x^3 + ax^2 + a^2x + a^3\).

This expanded polynomial is the original function broken down into its simplest components.
Expansion is crucial as it allows us to recognize the individual terms that we will later work with when dealing with derivatives, integrals, or series representations. Understanding polynomial expansion is an essential building block for more complex mathematical concepts like Taylor and Maclaurin series.

• It sets the stage for tackling higher-order mathematical problems by simplifying the initial challenge.

Finite Polynomial

A finite polynomial is a sum of a finite number of terms, each consisting of a constant multiplier times whole number powers of a variable. Such polynomials have distinct characteristics:

• True for any real number, they are easy to represent and calculate.
• Finite because they do not have an infinite series of terms.

For instance, the polynomial \(x^3 + ax^2 + a^2x + a^3\) derived from our original function is finite. Unlike infinite series that might require tests to determine convergence or divergence, finite polynomials automatically converge for all real numbers, as mentioned in the interval of convergence section.

• They offer certainty in evaluations and straightforward calculation of outputs for any input within the real number line.
• There are no conditions or restrictions on the domain, highlighting their advantages in practical solutions.

Understanding finite polynomials equips you to efficiently solve and analyze mathematical problems across various fields.

Series Representation

Series representation is a method of expressing a function as a sum of simpler component terms. In complex functions, we often transform these into series to simplify analysis and computation. Typically, series can be infinite or finite:

• Infinite series often need careful convergence analysis depending on the function.
• Finite series, like our polynomial \(x^3 + ax^2 + a^2x + a^3\), especially simple as their series form matches the polynomial itself.

Our given problem required converting the function into a power series. Since it involved a finite polynomial, its series representation naturally matched the polynomial form itself:

• \(x^3 + ax^2 + a^2x + a^3\).

This series representation facilitates much easier handling of mathematical computations and interpretations, especially when examining the behavior of functions across different values of \(x\). Understanding how to represent functions as series opens up methods to approximate, compute, and solve a broad range of calculus problems.