Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series (see Examples 1 and 2). $$ f(x)=\frac{1}{1+x} $$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Function Type

The function given is similar to a geometric series. Recall that the power series representation for \( \frac{1}{1-x} \) is \( \sum_{n=0}^{\infty} x^n \) with a radius of convergence of \( |x| < 1 \). To use this, we need to express \( f(x) \) in a similar form.

Step 2: Rewrite the Function

We have \( f(x) = \frac{1}{1+x} \). Rewrite it as \( f(x) = \frac{1}{1-(-x)} \). Now, the function is in the form \( \frac{1}{1-u} \) where \( u = -x \).

Step 3: Apply the Geometric Series Formula

Now that \( f(x) = \frac{1}{1-(-x)} \), we can say that \( f(x) = \sum_{n=0}^{\infty} (-x)^n \). This means that the power series representation is \( 1 - x + x^2 - x^3 + \cdots \).

Step 4: Determine the Radius of Convergence

From the geometric series \( \sum_{n=0}^{\infty} (-x)^n \), the radius of convergence is determined by \( |-x| < 1 \). This simplifies to \( |x| < 1 \). So, the radius of convergence is 1.

Key Concepts

Geometric Series

A geometric series is a series of the form \( a + ar + ar^2 + ar^3 + \ldots \). Here, \( a \) is the first term and \( r \) is the common ratio between the terms. It is called "geometric" because each term is found by multiplying the previous term by \( r \). Geometric series are extremely valuable when exploring power series, which allow complex functions to be expressed as infinite sums.

The sum of an infinite geometric series with \(|r| < 1\) is represented by:

• \( S = \frac{a}{1-r} \)

For a function like \( \frac{1}{1-x} \), it can be rewritten using a geometric series as \( \sum_{n=0}^{\infty} x^n \). This powerful formula is frequently used to manipulate functions into a form that is easily integrated or differentiated. Understanding geometric series is key in finding the power series representation for various functions, such as \( \frac{1}{1+x} \).

Radius of Convergence

The radius of convergence is an essential concept when dealing with power series. It indicates the interval around the center of convergence within which the series remains convergent.

To determine the radius of convergence, we look for the values of \( x \) for which the infinite sum converges. Mathematically, if \( a_n \) represents the coefficients of the series, the radius \( R \) can be determined by:

• \( R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \)

For the power series \( \sum_{n=0}^{\infty} (-x)^n \), the radius of convergence is derived from the condition \( |-x| < 1 \), which simplifies to \( |x| < 1 \). Therefore, the series converges for \( x \) within this interval. This implies that the series not only converges but also accurately represents the function for these values.

Convergent Series

A convergent series is a series in which the sum of an infinite number of terms approaches a finite limit. This is in contrast to divergent series, where the sum grows without bounds.

To verify if a series is convergent, tests like the Ratio Test, Root Test, or Comparison Test are often employed. For power series, convergence is typically determined within a radius of convergence. For example, the series \( \sum_{n=0}^{\infty} (-x)^n \) for \( f(x) = \frac{1}{1+x} \) converges within \( |x| < 1 \).

• Inside this interval, the series converges and represents the function.
• At the endpoints of the interval, further tests are necessary to determine convergence.

Understanding convergence helps us identify where a power series provides a valid representation of the function it approximates, crucial for applications in calculus and beyond.