Problem 10 - Thomas Calculus

Question

Find the first four terms of the binomial series for the functions. \begin{equation} \frac{x}{\sqrt[3]{1+x}} \end{equation}

Step-by-Step Solution

Answer

The first four terms are: \( x - \frac{1}{3}x^2 + \frac{2}{9}x^3 - \frac{7}{81}x^4 \).

Step 1: Express the Function Using Binomial Theorem

The given function is \( \frac{x}{(1+x)^{1/3}} \). We need to separate \( x \) from the binomial part. Here's how we will represent it:\[y = x (1+x)^{-1/3}\]This transforms the function into a form where we can apply the binomial theorem.

Step 2: Apply the Binomial Series Formula

The binomial series for \((1 + u)^n\) is given by:\[(1 + u)^n = 1 + nu + \frac{n(n-1)}{2!}u^2 + \frac{n(n-1)(n-2)}{3!}u^3 + \cdots \]For our function, let \( n = -\frac{1}{3} \) and \( u = x \).So the expansion becomes:\[(1+x)^{-1/3} = 1 - \frac{1}{3}x + \frac{(-1/3)(-4/3)}{2!}x^2 - \frac{(-1/3)(-4/3)(-7/3)}{3!}x^3 + \cdots \]

Step 3: Simplify the Binomial Series Terms

Now, we calculate the first few terms of the expansion:- The first term: \(1\)- The second term: \(-\frac{1}{3}x\)- The third term: \(\frac{(-1/3)(-4/3)}{2}x^2 = \frac{2}{9}x^2\)- The fourth term: \(\frac{(-1/3)(-4/3)(-7/3)}{6}x^3 = -\frac{7}{81}x^3\)

Step 4: Multiply by x to Get Each Term of the Series

We must multiply \( x \) by each of these simplified terms to find the expansion of the original expression:- Multiply each term by \( x \): - First term: \( x \) - Second term: \(-\frac{1}{3}x^2\) - Third term: \(\frac{2}{9}x^3\) - Fourth term: \(-\frac{7}{81}x^4\)

Step 5: Write Out the First Four Terms

The first four terms of the binomial series expansion for \( \frac{x}{(1+x)^{1/3}} \) are:\[ x - \frac{1}{3}x^2 + \frac{2}{9}x^3 - \frac{7}{81}x^4 \]

Key Concepts

Binomial Theorem

The binomial theorem provides a way to expand expressions that are raised to a power. Typically expressed as \((1 + u)^n\), it is a cornerstone of algebra and calculus. This theorem helps in turning complex expressions into a series where each term can be easily evaluated. It is especially useful when the exponent \(n\) is not an integer. This allows mathematicians to estimate functions and perform calculations even with fractional or negative exponents.

In our exercise, the binomial theorem is applied to expand \((1 + x)^{-1/3}\). This is an example of a case where the exponent is fractional. By employing the binomial theorem, we can express the function as a sum of terms, which makes it easier to evaluate when values of \(x\) are small.

Series Expansion

A series expansion represents a function as an infinite sum of simpler terms. It's often used to approximate functions that might otherwise be complex or cumbersome to handle. In our context, this involves taking the function \(\frac{x}{(1+x)^{1/3}}\) and writing it as a series.

By constructing it into a series, one can easily find the terms which contribute most significantly to the value of the function for small values of \(x\). Here, the first four terms were calculated to give a clear approximation:

• First Term: \(x\)
• Second Term: \(-\frac{1}{3}x^2\)
• Third Term: \(\frac{2}{9}x^3\)
• Fourth Term: \(-\frac{7}{81}x^4\)

These terms provide an approximation of the function, revealing how it behaves for small values of \(x\).

Mathematical Functions

Mathematical functions are expressions involving one or more variables. They model various phenomena in mathematics and other sciences. For the given exercise, understanding the function \(\frac{x}{(1+x)^{1/3}}\) is crucial for applying further mathematical operations to it.

Functions can often be transformed or expanded using different mathematical techniques such as the binomial theorem. This allows the expression to be manipulated and evaluated more easily. Recognizing patterns within these functions, especially factoring them into more manageable components, is essential in progressing through various mathematical problem-solving tasks.

Step-by-Step Solution

The step-by-step solution makes complex problems accessible. Following structured instructions, as shown in our exercise, enables individuals to systematically break down a problem.

First, the function was expressed in a suitable form for the binomial theorem. Then, the series expansion formula was applied to calculate each term separately. Finally, multiplying all terms by \(x\) provides the result. Each step refers back to previous knowledge, displaying the methodical approach needed in mathematical problem solving.This structured approach highlights the importance of understanding each part of the process, making sure each step is rooted in core mathematical principles.