Problem 3 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

1 + 3x + 6x^2 + 10x^3

Step 1: Understand the Binomial Series Expansion

The binomial series for a function of the form \((1 + x)^{n}\) is given by: \[\sum_{k=0}^{\infty} \binom{n}{k} x^k = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots\] In this problem, we are tasked with expanding \((1-x)^{-3}\). Let’s apply the series expansion with a specific focus on \(-x\) and \(n = -3\).

Step 2: Substitute Values into the Binomial Formula

Here, we substitute \(n = -3\) and \(x = -x\) into the binomial formula: \[(1-x)^{-3} = 1 + (-3)(-x) + \frac{(-3)(-3-1)}{2!}(-x)^2 + \frac{(-3)(-3-1)(-3-2)}{3!}(-x)^3 + \cdots\]

Step 3: Calculate Each Term

Using the formula, we calculate each of the first four terms individually:1. The first term is simply 1.2. The second term is \(-3(-x) = 3x\).3. The third term is \(\frac{(-3)(-4)}{2}(-x)^2 = \frac{12}{2}x^2 = 6x^2\).4. The fourth term is \(\frac{(-3)(-4)(-5)}{6}(-x)^3 = \frac{-60}{6}(-x)^3 = 10x^3\).

Step 4: Write the Series

Now we combine these terms to form the binomial series expansion for \((1-x)^{-3}\). Thus, the first four terms are:\[1 + 3x + 6x^2 + 10x^3\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that allows us to expand powers of expressions with two terms, or binomials. It defines the expansion of any power of a binomial in the form \((a + b)^n\). This theorem can be represented using a series, where each term involves binomial coefficients, denoted \(\binom{n}{k}\). These coefficients are calculated based on the factorial of numbers.

• The general formula for binomial expansion is: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
• In our exercise, we've used a variant of this series to expand \((1-x)^{-3}\), which involves a negative exponent.

Understanding the binomial theorem is essential as it is a stepping stone to more complex mathematical concepts.

Series Expansion

Series expansion involves representing functions as sums of terms generated from a specific formula. When we talk about binomial series expansion, it often involves using the binomial theorem where the series converges towards the original function's value as the number of terms increases.

In this context, we focused on expanding \((1-x)^{-3}\). The expansion is realized by working through each term of the series, utilizing the binomial formula for each successive power. This series expansion transforms complex expressions into a series of simpler terms. This is practical for calculations and understanding of how functions behave. You can expand many functions beyond simple binomials using series expansion techniques.

Negative Exponents

Negative exponents can seem tricky initially, but they follow simple rules. When you see an expression like \((1-x)^{-3}\), it denotes division. It is equivalent to \(\frac{1}{(1-x)^3}\).

• In binomial series context, a negative exponent indicates an infinite series expansion.
• The series for negative exponents doesn't terminate and continues as long as terms are needed for accuracy.
• The first few terms often give a good approximation for practical purposes.

In our example, considering a negative exponent in the binomial expansion allows us to compute the first few terms efficiently.

Power Series

A power series is an infinite series in the form:\[\sum_{k=0}^{\infty} a_k x^k\]Here, \(a_k\) represents the coefficients of the series, and \(x\) is a variable raised to successively higher powers.
For \((1-x)^{-3}\), we evaluate a power series using the binomial theorem. Each term in the series is composed of powers of \(x\) and the coefficients found through our series expansion. Understanding power series allows for complex functions to be analyzed in terms of simpler polynomial functions. Power series can solve equations that don't appear straightforward at first, especially for functions involving complex exponents or roots.