Problem 15 - Calculus: One and Several Variables

Question

Find a series expansion for the expression. $$\frac{x}{1-x} \text { for } | x,<1$$

Step-by-Step Solution

Answer

The power series representation for the expression \(\frac{x}{1-x}\) for |x| < 1 is: \[ \sum_{n = 1}^{\infty} x^{n} \]

Step 1: Identify the expression

We are given the function \(f(x) = \frac{x}{1-x}\), and we are tasked with finding a power series representation for it within the interval -1 < x < 1.

Step 2: Find power series representation

Recall the geometric series formula: \[ \sum_{n = 0}^{\infty} ar^{n} = \frac{a}{1-r} \] where r is the common ratio with |r| < 1. We want to get our expression in the form of \(\frac{a}{1-r}\), which is already given by \(f(x) = \frac{x}{1-x}\). So, we have: \[a = x \text{ and } r = x\] Since \(x \in (-1, 1)\), |r| = |x| < 1, which meets the convergence requirement.

Step 3: Write the power series representation

Now, we can write the power series representation for f(x) using the geometric series formula: \[ \sum_{n = 0}^{\infty} ar^{n} = f(x) = \frac{x}{1-x} \] Substitute the values of a and r: \[ \sum_{n = 0}^{\infty} (x)(x)^{n} = \sum_{n = 0}^{\infty} x^{n+1} = \sum_{n = 1}^{\infty} x^{n} \] The power series representation for the expression \(\frac{x}{1-x}\) for |x| < 1 is: \[ \sum_{n = 1}^{\infty} x^{n} \]

Key Concepts

Power Series Representation

Understanding the power series representation for a function is a cornerstone concept in calculus, particularly for functions that cannot easily be expressed using standard algebraic operations. A power series is an infinite series of the form \[ \sum_{n = 0}^{\text{\infty}} c_n (x - a)^n \] where \(c_n\) represents the coefficients, \(x\) is the variable, \(a\) is the center of the series, and \(n\) is a non-negative integer.

Using the power series, we can express complex functions as an infinite sum of simpler terms, allowing us to analyze and compute these functions more readily. In our given exercise, we needed to find the power series representation for the function \( f(x) = \frac{x}{1-x} \) within the interval \( -1 < x < 1 \). By identifying the function as a geometric series, we derived its power series representation, which is of practical use when approximating functions for calculations.

Geometric Series Formula

A geometric series is a series with a constant ratio between successive terms, which has a formula given by \[ \sum_{n = 0}^{\text{\infty}} ar^{n} = \frac{a}{1-r} \] where \( a \) is the first term, and \( r \) is the common ratio. This formula is valid for \( |r| < 1 \), due to the convergence requirement of the geometric series.

The equation \( \frac{x}{1-x} \) can be directly related to the formula of a geometric series, as exemplified in the exercise. The series expansion of \( \frac{x}{1-x} \) for \( |x| < 1 \) emerges naturally from the comparison, providing a ready-made formula for further analysis and application, such as computing limits, derivatives, and integrals of the function represented by the series.

Convergence of Series

In mathematics, the convergence of a series is paramount when working with infinite sequences and series. A series converges when the sum of its infinite terms approaches a finite limit. For power series, convergence is typically restricted to a specific interval or range of values, denoted as the radius of convergence.

In the case of the geometric series, convergence is guaranteed when the absolute value of the common ratio \( |r| \) is less than 1. This condition is vital to ensuring the series sum does not diverge to infinity. For \( f(x) = \frac{x}{1-x} \) given in the exercise, the series converges for \( |x| < 1 \) because the ratio \( r \) is the variable \( x \) itself, which lies within the convergence interval. Understanding the criteria for series convergence is not only crucial for ensuring series are valid but also provides insights into the behavior of functions represented by such series over different intervals.