Problem 31 - Calculus: Early Transcendentals

Question

Use the binomial series to expand the function as a power series. State the radius of convergence. \( \sqrt [4]{1 - x} \)

Step-by-Step Solution

Answer

Expansion: \(1 - \frac{1}{4}x + \frac{3}{32}x^2 - \frac{35}{384}x^3 + \ldots\); Radius: 1.

Step 1: Identify the Function to Expand

We start with the function \( f(x) = \sqrt[4]{1-x} \), which can be rewritten as \((1-x)^{1/4}\). Our goal is to expand this using the binomial series.

Step 2: Understand the Binomial Series Formula

The binomial series for \((1-x)^k\) where \(k\) is any real number is given by \[(1-x)^k = \sum_{n=0}^{\infty} \binom{k}{n} (-x)^n \]where \( \binom{k}{n} = \frac{k(k-1)(k-2)...(k-n+1)}{n!} \). We will apply this to \((1-x)^{1/4}\).

Step 3: Apply the Binomial Series

Substitute \(k = \frac{1}{4}\) into the formula:\[(1-x)^{1/4} = \sum_{n=0}^{\infty} \binom{1/4}{n} (-x)^n\] where the binomial coefficient is \(\binom{1/4}{n} = \frac{\frac{1}{4}((\frac{1}{4})-1)((\frac{1}{4})-2)...((\frac{1}{4})-n+1)}{n!}\).

Step 4: Simplify the First Few Terms

Calculate the first few terms using the formula:- For \(n=0\): \(\binom{1/4}{0}(-x)^0 = 1\)- For \(n=1\): \(\binom{1/4}{1}(-x) = \frac{1}{4}(-x)\)- For \(n=2\): \(\binom{1/4}{2}(-x)^2 = \frac{1}{4} \times -\frac{3}{4} \times \frac{x^2}{2} = \frac{3x^2}{32}\)- For \(n=3\): \(\binom{1/4}{3}(-x)^3 = \frac{1}{4} \times -\frac{3}{4} \times -\frac{7}{4} \times \frac{x^3}{6} = -\frac{35x^3}{384}\)

Step 5: Write the Series Expansion

The power series expansion of \((1-x)^{1/4}\) is:\[1 - \frac{1}{4}x + \frac{3}{32}x^2 - \frac{35}{384}x^3 + \ldots\] This is the binomial series expansion of \(\sqrt[4]{1-x}\).

Step 6: Determine the Radius of Convergence

For the binomial series \((1-x)^k\) with \(k\) as any real number, the radius of convergence is \(R = 1\). Therefore, the radius of convergence for \((1-x)^{1/4}\) is 1.

Key Concepts

Power Series Expansion

Power series expansion is a method used to represent functions as an infinite sum of terms. These terms are derived from the function evaluated at a certain point, typically a point where the series converges. In this exercise, we expanded the function \(\sqrt [4]{1-x}\), which was rewritten as \((1-x)^{1/4}\). This transformation allows us to apply the binomial series formula.

The general form for expanding \((1-x)^k\) into a power series is given by:

• \((1-x)^k = \sum_{n=0}^{\infty} \binom{k}{n} (-x)^n \)

Here, each term of the series involves a binomial coefficient and increasingly higher powers of \(-x\).

This approach makes it easier to analyze functions and approximate them near a chosen point. By calculating the first few terms, you can get a good approximation of the function. For instance, the series \(1 - \frac{1}{4}x + \frac{3}{32}x^2 - \frac{35}{384}x^3 + \ldots\) gives an idea of how \(\sqrt [4]{1-x}\) behaves around \(x=0\). The series allows us to easily compute values and understand the behavior of the function at different points.

Radius of Convergence

The radius of convergence is a crucial concept when dealing with power series. It defines the interval within which a power series converges to the function it represents. Ensuring that a series converges is vital because outside this radius, the series may not approximate the function correctly.

For a binomial series like \((1-x)^k\) with any real \(k\), the radius of convergence \(R\) is 1. This means that the series accurately represents the function on the interval \(-1 < x < 1\).

How do we determine this? The binomial series is designed to ensure convergence on such intervals. By setting \(x\) values within \(-1\) to \(1\), we know that the sum of infinite terms will approach a real number, capturing the behavior of \((1-x)^{1/4}\). This property helps in approximating values accurately and analyzing the function's behavior both visually and computationally within the valid interval.

Binomial Coefficients

Binomial coefficients are the backbone of binomial expansions, acting as the multipliers for each term in the series. When expanding \((1-x)^{k}\), the binomial coefficient for the term \(n\) is represented as \(\binom{k}{n}\). It is given by the formula:

• \(\binom{k}{n} = \frac{k(k-1)(k-2)...(k-n+1)}{n!}\)

This formula generalizes the idea of combinations for any real number \(k\), allowing us to expand functions with non-integer powers like \((1-x)^{1/4}\).

In our example, for \(n=0,1,2,3\), the configured coefficients calculated are \(1\), \(\frac{1}{4}\), \(\frac{3}{32}\), and \(-\frac{35}{384}\), respectively. With each term in the series, these coefficients scale the power of \(-x\) derived from the expansion.

Understanding these coefficients is essential for grasping how the series builds up to match the function. As the terms increase, the complexity of the coefficient grows, influencing the contribution of higher-degree terms. This behavior ensures that the series models the function accurately over its radius of convergence.