Problem 12 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The binomial series is \( 1 + 3x^2 + 3x^4 + x^6 \).

Step 1: Identify Binomial Series Formula

The binomial series for \[ (1 + x)^n \] is given by:\[ \sum_{k=0}^{n} \binom{n}{k} x^k = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots \]where \( \binom{n}{k} \) is the binomial coefficient.

Step 2: Set Up for \( (1 + x^2)^3 \)

To find the binomial series for the function \( (1 + x^2)^3 \), we'll apply the formula for \((1 + x)^n\) by replacing \(x\) with \(x^2\). Here, \(n = 3\).

Step 3: Calculate Binomial Coefficients

Calculate the binomial coefficients for \((1 + x^2)^3\):\[ \binom{3}{0} = 1, \quad \binom{3}{1} = 3, \quad \binom{3}{2} = 3, \quad \binom{3}{3} = 1 \].

Step 4: Expand the Series

Expand the series using the binomial coefficients and the power of \(x^2\):\[ (1 + x^2)^3 = \binom{3}{0}(x^2)^0 + \binom{3}{1}(x^2)^1 + \binom{3}{2}(x^2)^2 + \binom{3}{3}(x^2)^3 \].

Step 5: Simplify Each Term

Simplify each term in the expansion:\[ 1 \cdot 1 + 3 \cdot x^2 + 3 \cdot (x^2)^2 + 1 \cdot (x^2)^3 = 1 + 3x^2 + 3x^4 + x^6 \].

Step 6: Write the Final Series

The binomial series expansion of \((1+x^2)^3\) is:\[ 1 + 3x^2 + 3x^4 + x^6 \].

Key Concepts

Binomial Coefficients

When working with binomial expansions, the term "binomial coefficients" refers to the numbers that serve as the coefficients for each term in the expansion. These are expressed using the symbol \( \binom{n}{k} \). Sometimes, you might hear them called "combinations" because they represent the number of ways to choose \(k\) items from \(n\) without regard to order.

In formula terms, a binomial coefficient is calculated as follows:

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

Where \(!\) denotes a factorial, meaning you multiply the number by all the integers below it until you reach 1. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In our example of \( (1 + x^2)^3 \), the binomial coefficients are:

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

Understanding these coefficients is crucial because they tell us how much each term in the expansion "weighs." Each coefficient shows up as a multiplier for the term it accompanies in the expansion, greatly influencing the final output of the series.

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial expression as a sum of terms. The terms are derived from powering up each piece of the original binomial and multiplying them by the appropriate binomial coefficients.

For instance, when you deal with \((1 + x^2)^3\), you're involving polynomial expansion to translate this power into a sum of individual terms. In this exercise, we replace our variable \( x \) with \( x^2 \) in the binomial template \( (1 + x)^3 \). Recognizing and substituting this is crucial. Each power of \( x^2 \) is multiplied by its corresponding coefficient:

• Start with \( 1 \cdot (x^2)^0 = 1 \)
• Then add \( 3 \cdot (x^2)^1 = 3x^2 \)
• Continue to \( 3 \cdot (x^2)^2 = 3x^4 \)
• Finally, \( 1 \cdot (x^2)^3 = x^6 \)

Together, all these steps "expand" the polynomial so you end up with a series that represents the original function neatly as a sum of powers of \( x^2 \).

By seeing this detailed breakdown, each term soldiers a logical structural flow from the base polynomial form, making it much easier to understand how expansions are executed.

Series Expansion

A series expansion is a powerful mathematical technique that allows you to express a complicated function in terms of an infinite sum of simpler terms. For polynomials and some other types of functions, however, the series often terminates after a finite number of terms, turning it into a simpler and finite expression.

In the exercise for \( (1 + x^2)^3 \), the series expansion isn't infinite because it's a polynomial. You decide on the highest power to expand to, in this case, \(x^6\), based on the exponent \(n = 3\) of the original expression. Even in finite cases like polynomials, understanding the "series" mindset allows you to visualize how each term contributes to the overarching result.

• Each \( \binom{n}{k} \) helps sculpt the "shape" of the series.
• Powers of the binomial and resulting terms are added together as a sequence.
• Even though the series in this case is "finite," the process gives insights applicable to broader mathematical analysis.

In summary, through the series expansion, you transform the expression \( (1 + x^2)^3 \) into tangible terms you can work with: \( 1 + 3x^2 + 3x^4 + x^6 \), effectively simplifying complex mathematical functions into more manageable pieces. This concept extends not just to binomials but to various complex functions in calculus and beyond.