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

The first four terms are: 1, -3x, 6x^2, -10x^3.

Step 1: Identify the Binomial Series Formula

The binomial series \[(1-x)^n = \sum_{k=0}^{\infty} \binom{n}{k} (-x)^k\] is used when \(|x| < 1\) and \(n\) is any real number.

Step 2: Substitute and Calculate Binomial Coefficients

For the function \((1-x)^{-3}\), we substitute \(n = -3\) into the binomial series formula.The general term in the series is \[\binom{-3}{k} = \frac{(-3)(-3-1)(-3-2)...(-3-k+1)}{k!}.\] Calculate the first four binomial coefficients for \(k = 0, 1, 2, 3\).

Step 3: Calculate the First Term (\(k = 0\))

The coefficient for \(k = 0\) is \[\binom{-3}{0} = 1.\] So, the first term is \[1.\]

Step 4: Calculate the Second Term (\(k = 1\))

For \(k = 1\), the coefficient is \[\binom{-3}{1} = -3.\] Therefore, the second term is\[-3x.\]

Step 5: Calculate the Third Term (\(k = 2\))

For \(k = 2\), the coefficient \[\binom{-3}{2} = \frac{-3 \cdot (-4)}{2!} = 6.\] Thus, the third term is\[6x^2.\]

Step 6: Calculate the Fourth Term (\(k = 3\))

For \(k = 3\), the coefficient \[\binom{-3}{3} = \frac{-3 \cdot (-4) \cdot (-5)}{3!} = -10.\] So, the fourth term is\[-10x^3.\]

Step 7: Combine the First Four Terms

Combining all the terms calculated together, the first four terms of the series are:\[1 - 3x + 6x^2 - 10x^3.\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a cornerstone in algebra that allows us to expand expressions of the form \((1+x)^n\). It provides a way to express these powers as sums of terms involving binomial coefficients. Each term in the series has the format \(\binom{n}{k} x^k\), where \(\binom{n}{k}\) are the binomial coefficients. This theorem is particularly powerful because it works not only for positive integers but also for any real number \(n\) when using the binomial series expansion:
\[(1+x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k\].

Some key applications and characteristics are:

• When \(n\) is a positive integer, the series becomes finite, making it a simple polynomial.
• For other real values, the expansion continues infinitely as a series, valid for \(|x| < 1\).

The theorem simplifies calculations in algebra, calculus, and can even help in evaluating limits and other complex expressions in mathematical analysis.

Binomial Coefficients

Binomial Coefficients are denoted as \(\binom{n}{k}\) and represent the coefficients in the binomial expansion. They appear in each term of a binomial series and hold special mathematical significance. The formula for binomial coefficients is:
\[\binom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!}\],
where \(k!\) is the factorial of \(k\). Understanding these coefficients is key to applying the Binomial Theorem, and they have unique properties:

• Symmetry Property: \(\binom{n}{k} = \binom{n}{n-k}\), which simplifies calculations.
• Binomial coefficients count the number of ways to choose \(k\) items from \(n\) without regard to order, making them significant in combinatorics.
• They are integral to Pascal's Triangle, a convenient tool for remembering binomial coefficients quickly.

By mastering these coefficients, you will be able to expand binomials easily and understand the intricacies of infinite series expansions involving them.

Infinite Series

An Infinite Series represents a sum of infinitely many terms. In the context of the Binomial Theorem, it describes how a power of a binomial can be expanded into an endless sequence of terms. This is applicable when \(n\) is not a non-negative integer, resulting in an expansion valid within a certain radius of convergence, \(|x| < 1\).

Key points about infinite series include:

• An infinite series typically has no final term, though an initial portion can approximate the value.
• Convergence is crucial; the sum approaches a fixed value as more terms are added, so understanding the conditions for convergence is important.
• In practical use, only a finite number of initial terms may be calculated (like in our binomial series example, finding the first four terms).

Infinite series are foundational in calculus and analysis, providing frameworks to solve complex problems, compute approximations, and express functions in series form. Studying them will deepen your understanding of mathematical expansions and convergence behavior.