Problem 13 - Thomas Calculus

Question

Find the binomial series for the functions. \begin{equation} (1-2 x)^{3} \end{equation}

Step-by-Step Solution

Answer

The binomial series is \((1-2x)^3 = 1 - 6x + 12x^2 - 8x^3.\)

Step 1: Understand the Binomial Series Formula

The binomial series expansion formula for a function \((1+x)^n\) is \( \sum_{k=0}^{\infty} {n \choose k} x^k \), where \({n \choose k} = \frac{n!}{k!(n-k)!}\)\ is the binomial coefficient. Our aim is to find the expansion of \((1-2x)^3\).

Step 2: Substitute Variables and Identify the Pattern

In this problem, \((1-2x)^3\), we can set \(n=3\) and identify that \(x\) is replaced by \(-2x\). The series then becomes \( \sum_{k=0}^{\infty} {3 \choose k} (-2x)^k \).

Step 3: Calculate the Binomial Coefficients

Calculate the binomial coefficients for \(k = 0, 1, 2, 3\): - \({3 \choose 0} = 1\)- \({3 \choose 1} = 3\)- \({3 \choose 2} = 3\)- \({3 \choose 3} = 1\).

Step 4: Evaluate Each Term of the Series

For each \(k\):- For \(k=0\): \({3 \choose 0}(-2x)^0 = 1\)- For \(k=1\): \({3 \choose 1}(-2x)^1 = 3(-2x) = -6x\)- For \(k=2\): \({3 \choose 2}(-2x)^2 = 3(4x^2) = 12x^2\)- For \(k=3\): \({3 \choose 3}(-2x)^3 = 1(-8x^3) = -8x^3\).

Step 5: Write the Series Expansion

Combining all the evaluated terms, the binomial series for \((1-2x)^3\) is given by:\[(1-2x)^3 = 1 - 6x + 12x^2 - 8x^3.\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power. It applies specifically to binomial expressions, which are expressions made up of two terms.

The general form of the Binomial Theorem is given as:

• \( (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k} \)

The terms in these expansions are determined by binomial coefficients and follow a specific pattern. For example, the expansion of \((1+x)^n\) using the binomial theorem results in:

• \( \sum_{k=0}^{n} {n \choose k} x^k \)

This theorem is useful for simplifying calculations and understanding patterns in higher powers of binomials.

Series Expansion

Series expansion involves expressing a function as an infinite sum of terms. It is extremely helpful in approximating functions that are otherwise difficult to handle in their given form.

In the context of the Binomial Series, series expansion lets us represent

• \((1-2x)^3\) as \( \sum_{k=0}^{\infty} {3 \choose k} (-2x)^k \)

The function \((1-2x)^3\) can be expanded into a series of terms by calculating each term individually. This allows us to evaluate complex expressions easily and is particularly important in calculus and numerical methods.

Binomial Coefficient

The binomial coefficient is an integral part of the Binomial Theorem and provides the coefficients of each term in the expansion. It is denoted as

• \( {n \choose k} = \frac{n!}{k!(n-k)!} \)

The binomial coefficient represents the number of ways to choose \(k\) elements from a set of \(n\) elements and determines the weight of each term in the expansion.

For example, in expanding\((1-2x)^3\), we calculate binomial coefficients for different values of \(k\):

• \({3 \choose 0} = 1\)
• \({3 \choose 1} = 3\)
• \({3 \choose 2} = 3\)
• \({3 \choose 3} = 1\)

These coefficients help in building the series \((1-2x)^3 = 1 - 6x + 12x^2 - 8x^3\), showing how each term in the series is constructed.