Problem 24 - Essential Calculus Early Transcendentals

Question

Use the binomial series to expand the function as a power serics. State the radius of convergence. $$\sqrt[3]{8+x}$$

Step-by-Step Solution

Answer

The power series is \( 2 (1 + \frac{x}{8})^{1/3} \). Radius of convergence: 8.

Step 1: Identify the Binomial Series Formula

The binomial series for \( (1 + x)^n \) is given by \(= \sum_{k=0}^{\infty} \binom{n}{k} x^k\\), where \( \binom{n}{k} = \frac{n(n-1)(n-2)\dots(n-k+1)}{k!} \). Here, we need to transform \( (8+x)^{1/3} \) into the form \( (1 + u)^n \).

Step 2: Rewrite Function into Binomial Form

We start by rewriting \( \sqrt[3]{8+x} = (8+x)^{1/3} \) as \( 2\left(1 + \frac{x}{8}\right)^{1/3} \). We can now focus on expanding \( \left(1 + \frac{x}{8}\right)^{1/3} \) using the binomial series.

Step 3: Apply the Binomial Series Expansion

Using the binomial series formula, we have:- Coefficient of each term determined by \( \binom{1/3}{k} = \frac{\frac{1}{3} \cdot (\frac{1}{3} - 1) \cdot \dots \cdot (\frac{1}{3} - k + 1)}{k!} \).- Expansion becomes: \( \sum_{k=0}^{\infty} \binom{1/3}{k} \left(\frac{x}{8}\right)^k \).- Calculate first few terms to observe the pattern.

Step 4: Calculate First Few Terms

For \( k=0 \), term is \( 1 \).For \( k=1 \), term is \( \frac{1}{3} \cdot \frac{x}{8} = \frac{x}{24} \).For \( k=2 \), term is \( \frac{1}{3} \cdot \left(-\frac{2}{3}\right) \cdot \frac{1}{2!} \cdot \left(\frac{x}{8}\right)^2 = -\frac{x^2}{576} \).Continue this process to find more terms.

Step 5: Determine the Radius of Convergence

The series \( \left(1 + \frac{x}{8}\right)^{1/3} \) converges for \( \left| \frac{x}{8} \right| < 1 \), which simplifies to \( |x| < 8 \). Thus, the radius of convergence is 8.

Key Concepts

Power Series Expansion

The concept of a power series expansion is fundamental in expressing functions as infinite sums of terms. It essentially breaks down complex functions into infinite series that are often easier to manage and analyze. In general, a power series centered at some point can be written as:
\[ \sum_{k=0}^{\infty} a_k (x-c)^k\]where:

• \(a_k\) represents the coefficient of each term.
• \(c\) is the center of the series, often zero in basic series.

In our original exercise involving \(\sqrt[3]{8+x}\), we expressed it in a form amenable to power series expansion. Rewriting \((8+x)^{1/3}\) as \(2(1 + \frac{x}{8})^{1/3}\) allows us to utilize the binomial series formula. This transformation is crucial because it simplifies the expansion by turning it into a familiar and manageable format.
Engaging with power series helps in approximating functions over specific intervals, and such expressions often converge to functions only within certain bounds, analyzed via the radius of convergence.

Radius of Convergence

The radius of convergence is a critical property of power series that tells us about the interval within which the series converges to a function. It is important because it determines the range of \(x\) values for which our series expansion is valid and reliable. Mathematically, for a power series \(\sum_{k=0}^{\infty} a_k (x-c)^k\), the radius of convergence \(R\) can be found using formulas derived from the Ratio Test or the Root Test.
In the context of our exercise, we needed to find where the series \((1 + \frac{x}{8})^{1/3}\) converges. The often-used formula,

• \(\left| \frac{x}{8} \right| < 1\), simplifies to \(|x| < 8\),

gives us a radius of convergence of 8.
This means our series expansion of the function \(\sqrt[3]{8+x}\) using binomial expansion is valid for \(x\) values within the interval (-8, 8). This critical boundary helps prevent misapplication of series outside intended limits.

Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions that are raised to any power, particularly fractional or negative ones. It provides a systematic way to expand \((1 + x)^n\) into a power series and is especially useful when \(n\) is not an integer. The binomial theorem formula is:
\[ (1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k \]where:

• \(\binom{n}{k} = \frac{n(n-1)(n-2)\dots(n-k+1)}{k!}\) is the generalized binomial coefficient.

In our exercise, we used this theorem to convert \( (8+x)^{1/3} \) into a series by rewriting it as \( 2(1 + \frac{x}{8})^{1/3} \). This allowed us to apply the formula:

• The coefficient of \(x^k\) in the series is determined by \(\binom{1/3}{k}\).
• This involves calculating factors like \(\binom{1/3}{0}\), \(\binom{1/3}{1}\), \(\binom{1/3}{2}\), and so on.

Each term's coefficient involves multiple calculations, illustrating the methodical approach required when using the binomial theorem for non-integer powers.