Problem 7 - Essential Calculus Early Transcendentals

Question

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{x}{9+x^{2}} $$

Step-by-Step Solution

Answer

The power series is \(\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{9^{n+1}}\) with interval of convergence \((-3, 3)\).

Step 1: Identify the Form

We start by identifying a known form for which we can find a power series representation. The function \( f(x) = \frac{x}{9 + x^2} \) can be decomposed such that \( \frac{1}{9 + x^2} \) resembles a geometric series when rewritten.

Step 2: Rewrite the Denominator

Converting the denominator to a form resembling \( \frac{1}{1-u} \) gives us: \( \frac{1}{9 + x^2} = \frac{1}{9(1 + \frac{x^2}{9})} = \frac{1}{9} \cdot \frac{1}{1 - (-\frac{x^2}{9})} \). This sets us up to use the geometric series.

Step 3: Apply Geometric Series Expansion

Using the geometric series \( \frac{1}{1-u} = \sum_{n=0}^{\infty} u^n \), substitute \( u = -\frac{x^2}{9} \) to get \( \frac{1}{1 - (-\frac{x^2}{9})} = \sum_{n=0}^{\infty} \left(-\frac{x^2}{9}\right)^n \).

Step 4: Incorporate the Original Function

Multiply the series from Step 3 by \( \frac{x}{9} \) to match \( f(x) \): \[\frac{x}{9} \cdot \sum_{n=0}^{\infty} \left(-\frac{x^2}{9}\right)^n = \sum_{n=0}^{\infty} \frac{x}{9} \left(-\frac{x^{2n}}{9^n}\right) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{9^{n+1}} \].

Step 5: Determine Interval of Convergence

For the geometric series \( |u| < 1 \) must hold for convergence. In this case, \( |-\frac{x^2}{9}| < 1 \), simplifying to \( |x^2| < 9 \), so \( |x| < 3 \). Thus, the interval of convergence is \((-3, 3)\).

Key Concepts

Interval of Convergence

When dealing with power series, understanding the interval of convergence is crucial. This interval tells us for which values of the variable, say \( x \), the series will converge to a sum. Think of it as the "safe zone" for our series where it behaves and gives meaningful results.

For a geometric series, the convergence condition is typically \( |u| < 1 \), where \( u \) is the common ratio. Applied to our original function, we have the transformed ratio \( -\frac{x^2}{9} \).

Therefore, ensuring convergence leads us to:

• Condition: \(|-\frac{x^2}{9}| < 1\)
• Which simplifies to \(|x^2| < 9\)
• Or even further to \(|x| < 3\)

This means our interval of convergence, where the series representation holds true, is \((-3, 3)\). Understanding this ensures we use the power series properly within its limits.

Geometric Series

A geometric series is a one of the simplest types of series and is very handy in calculus. It's characterized by a constant ratio between consecutive terms. The general form is \( \frac{1}{1-u} = \sum_{n=0}^{\infty} u^n \), making it very approachable for manipulation.

In practical terms, if you have any expression that can be rewritten to match \( \frac{1}{1-u} \), you can easily expand it into a series. This makes geometric series a powerful tool in finding approximations of functions.

In our exercise, the original function \( \frac{1}{9 + x^2} \) was adjusted to fit this model by transforming it to \( \frac{1}{9} \cdot \frac{1}{1 - (-\frac{x^2}{9})} \). This transformation allowed us to apply the geometric series expansion and further simplify calculations.

Series Representation

Series representation is a method used in calculus to express functions as an infinite sum of terms based on a specific variable, often \( x \). This helps in approximating and understanding functions better, especially those that are complex or non-linear.

The function \( f(x) = \frac{x}{9 + x^2} \) found its series representation using the concept of "borrowed" geometric series. By transforming part of the function to suit the format of a geometric series, we extracted a power series representation:

• The transformation produced \( \frac{x}{9} \cdot \sum_{n=0}^{\infty} \left(-\frac{x^2}{9}\right)^n \)
• This simplified into: \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{9^{n+1}} \)

Such a representation allows for easier integration, differentiation, and evaluation of functions within the interval of convergence. It also provides valuable insight into the behavior of the function across different values of \( x \).