Problem 6 - Thomas Calculus

Question

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1-\frac{x}{3}\right)^{4} \end{equation}

Step-by-Step Solution

Answer

The first four terms are \(1 - \frac{4x}{3} + \frac{2x^2}{3} - \frac{4x^3}{27}\).

Step 1: Identify the Binomial Series Formula

The binomial series for the expansion of \((1 - x)^n\) is given by \((1 - x)^n = 1 - nx + \frac{n(n-1)}{2!}x^2 - \frac{n(n-1)(n-2)}{3!}x^3 + \ldots\). The pattern continues with alternating signs.

Step 2: Plug in the Values

In this exercise, we have \((1 - \frac{x}{3})^4\). This can be written as \((1 - u)^4\) where \(u = \frac{x}{3}\). Substitute \(u = \frac{x}{3}\) and \(n = 4\) into the binomial series formula.

Step 3: Calculate Each Term of the Series

Calculate the first four terms using the values:1. First term: \(1\)2. Second term: \(-4 \cdot \frac{x}{3}\)3. Third term: \(6 \cdot \left(\frac{x}{3}\right)^2 = 6 \cdot \frac{x^2}{9}\)4. Fourth term: \(-4 \cdot 3 \cdot \left(\frac{x}{3}\right)^3 = -4 \cdot \frac{x^3}{27}\)

Step 4: Simplify the Terms

Simplify each term:1. First term: \(1\)2. Second term: \(-\frac{4x}{3}\)3. Third term: \(\frac{2x^2}{3}\)4. Fourth term: \(-\frac{4x^3}{27}\)

Step 5: Write the First Four Terms of the Binomial Series

Combine the simplified terms to write the first four terms of the expansion, resulting in: \(1 - \frac{4x}{3} + \frac{2x^2}{3} - \frac{4x^3}{27}\).

Key Concepts

Binomial Theorem

The binomial theorem is an important formula in algebra that helps us expand expressions that are raised to a power. For any positive integer \( n \), it states: \[ (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \] This theorem can be used even when \( n \) is not an integer, like when dealing with fractions or decimals. It gives us an infinite series where the coefficients are determined by the binomial coefficients, which are the numbers that appear in Pascal's triangle.

• The terms alternate between positive and negative when the original expression has a negative sign, as seen in the problem \( (1-\frac{x}{3})^4 \).
• The power of each term grows sequentially.
• The binomial coefficients are calculated using combinations: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

These coefficients provide insight into how many ways certain items can be chosen, a concept fundamental in combinatorics.

Binomial Expansion

Binomial expansion uses the binomial theorem to expand expressions raised to a power, such as \( (1 - \frac{x}{3})^4 \). The expansion is a way of expressing a binomial raised to a power as a sum of terms in the form of a polynomial. In the given example:

• Rewrite it to match the binomial form: \( (1 - u)^4 \), where \( u = \frac{x}{3} \).
• The first 4 terms are calculated using the expansion: \ 1 - 4u + 6u^2 - 4u^3 \.
• Substitute \( u \) again to return to \( x \): this results in terms like \( -\frac{4x}{3} \) and \( -\frac{4x^3}{27} \).

By following these specific steps, you can generate a simplified polynomial. Each term gives a piece of how the binomial grows, with higher powers having diminishing impact thanks to the division by increasingly large factorials.

Calculus

Calculus often intersects with the binomial theorem when exploring series expansions. Understanding the binomial series is crucial for grasping more complex topics like Taylor and Maclaurin series, which are essential in calculus.

• Binomial series can approximate functions over a small range, especially using derivatives evaluated at a point.
• These expansions help solve calculus problems by representing difficult functions in simpler polynomial forms.

The series expansion results derived from the binomial theorem serve as approximations that can simplify integration or differentiation of expressions. Such techniques are powerful, allowing intricate expressions to become manageable, which is a key strength of calculus in analyzing functions deeply.