Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series. $$ f(x)=\frac{1}{1+x} $$

Step-by-Step Solution

Answer

The power series representation is \( \sum_{n=0}^{\infty} (-x)^n \) with a radius of convergence of 1.

Step 1: Recognize the Geometric Series Form

The function given is similar to the form of a geometric series. A geometric series is generally given by \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \) for \( |x| < 1\). We should try to rewrite \( f(x) = \frac{1}{1+x} \) in a form that resembles this geometric series.

Step 2: Rewrite the Function

Notice that we can express the denominator as \( 1 - (-x) \). Thus, we rewrite the function as \( f(x) = \frac{1}{1-(-x)} \). This matches the geometric series form \( \frac{1}{1-x} \), allowing us to use the series representation.

Step 3: Determine the Power Series Representation

Using the geometric series formula, substitute \(-x\) for \(x\) in \( \sum_{n=0}^{\infty} x^n \). This gives:\[\sum_{n=0}^{\infty} (-x)^n = 1 - x + x^2 - x^3 + \ldots\]This is the power series representation for \( f(x) = \frac{1}{1+x} \).

Step 4: Calculate the Radius of Convergence

The radius of convergence for a geometric series \( \frac{1}{1-x} \) is \( |x| < 1 \). Here, we have \( | -x | < 1 \), which simplifies to \( |x| < 1 \). Therefore, the radius of convergence is 1 for this power series.

Key Concepts

Understanding Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This series has a very particular form. It's expressed as \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \) for \( |x| < 1\). This formula provides the sum of all the terms when \(x\) remains within the range for convergence.
The concept of a geometric series is integral in creating power series from elementary functions.

• The first term in a geometric series is always 1.
• Each subsequent term is derived by multiplying the common factor \(x\) by the previous term.
• The series continues infinitely, but only converges if \( |x| < 1\) .

Recognizing a geometric series helps when transforming functions into their corresponding series representation. For instance, to express \( f(x) = \frac{1}{1+x} \) in this form, you rewrite it to match the geometric series setup. In this case, you observe it as \( \frac{1}{1-(-x)} \), allowing us to view \(-x\) as the common ratio
of the geometric series.

Exploring the Radius of Convergence

When working with series, it's crucial to determine its radius of convergence. This tells us the range of \(x\) values for which the series converges to a finite sum. Think of this as the interval within which, whatever terms you calculate, the sum will be settled and well-behaved.
The radius of convergence for a geometric series \( \frac{1}{1-x} \) is \( |x| < 1\) . For our series \( \sum_{n=0}^{\infty} (-x)^n \), this results in the inequality \( |-x| < 1 \). Since absolute values look at magnitude only, this simplifies cleanly to \( |x| < 1 \).

• Within this radius, the series converges to an exact value.
• Outside this radius, the series diverges, meaning it doesn't settle on a specific value.
• The calculation of convergence is essential for the accurate representation of any function as a power series.

Understanding convergence helps in knowing where the series representation of a function accurately replicates its values. In practical terms, it means we can trust the series-based calculations within the given range.

Series Representation Basics

Series representation is a method for expressing functions as an infinite sum of terms from a sequence. It's like peeling back layers to see a function's components; each layer helps to build the complete picture. Power series allow us to represent complex functions in terms of a simple geometric series.
To arrive at a series representation for \( f(x) = \frac{1}{1+x} \), we start by recognizing its link to a geometric series. By rewriting it as \( \frac{1}{1-(-x)} \), it fits perfectly into the formula \( \sum_{n=0}^{\infty} (-x)^n \). This means:

• \(1\) when \(n = 0\),
• \(-x\) when \(n = 1\),
• \(x^2\) when \(n = 2\), and so on.

Each term becomes a building block, combining to form an accurate portrayal of \(f(x)\). Power series are invaluable in various fields, providing essential methods for analyzing functions, especially when direct calculations are cumbersome.
Mastering series representation gives you powerful tools to tackle even the most complex functions, paving the way for deeper mathematical understanding.