Problem 14 - Thomas Calculus

Question

Find the binomial series for the functions. \begin{equation} \left(1-\frac{x}{2}\right)^{4} \end{equation}

Step-by-Step Solution

Answer

The binomial series for \((1 - \frac{x}{2})^4\) is \(1 - 2x + \frac{3x^2}{2} - \frac{x^3}{2} + \frac{x^4}{16}\).

Step 1: Identify the Binomial Series

To find the binomial series for \( (1 - \frac{x}{2})^4 \), we need to recall the binomial theorem. The binomial theorem states that for any real number \( n \) and \( |x| < 1 \),\[(1 + x)^{n} = \sum_{k=0}^{\infty} \binom{n}{k} x^k\]Here, \( n = 4 \) and \( x \) in our equation is replaced with \(-\frac{x}{2}\).

Step 2: Substitute Values

Using the binomial theorem:\[(1 - \frac{x}{2})^4 = \sum_{k=0}^{\infty} \binom{4}{k} \left(-\frac{x}{2}\right)^k\]This series will expand using the formula for binomial coefficients.

Step 3: Calculate Binomial Coefficients

Calculate the binomial coefficients \( \binom{4}{k} \) for the first few terms, starting from \( k = 0 \) to \( k = 4 \) (since terms beyond \( k = 4 \) will vanish as they involve division by zero or powers leading to zero):- \( \binom{4}{0} = 1 \)- \( \binom{4}{1} = 4 \)- \( \binom{4}{2} = 6 \)- \( \binom{4}{3} = 4 \)- \( \binom{4}{4} = 1 \)

Step 4: Compute Each Term

Substitute each binomial coefficient and compute each term:- For \( k = 0 \): \( 1 \times (1) = 1 \)- For \( k = 1 \): \( 4 \times (-\frac{x}{2}) = -2x \)- For \( k = 2 \): \( 6 \times \left(-\frac{x}{2}\right)^2 = \frac{3x^2}{2} \)- For \( k = 3 \): \( 4 \times \left(-\frac{x}{2}\right)^3 = -\frac{x^3}{2} \)- For \( k = 4 \): \( 1 \times \left(-\frac{x}{2}\right)^4 = \frac{x^4}{16} \)

Step 5: Write the Binomial Series

Combine the terms to write the binomial series expansion for \( (1 - \frac{x}{2})^4 \):\[1 - 2x + \frac{3x^2}{2} - \frac{x^3}{2} + \frac{x^4}{16}\]

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept in algebra that describes the algebraic expansion of powers of a binomial. A binomial is an expression that consists of two terms, like \((a + b)\). The theorem states that for any real number \(n\), the expansion of \((a + b)^n\) can be expressed as a sum involving binomial coefficients. This is given by:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In simple terms, it allows one to expand powers of any sum of two terms into a series or sum of terms involving coefficients. Each term in the expansion contains a binomial coefficient, the first term \(a\) raised to a decreasing power, and the second term \(b\) raised to an increasing power.

• It's widely used in algebra, calculus, and analysis.
• The series terminates at \(n\), which is a finite series if \(n\) is a non-negative integer.
• Provides insights into polynomial expansion and combinatorial arrangements.

Understanding this theorem is key as it lays the groundwork for further concepts in mathematics, such as calculus and combinatorics.

Binomial Coefficients

Binomial coefficients are crucial in the binomial theorem expansion. They are represented by the symbol \(\binom{n}{k}\), often read as "\(n\) choose \(k\)," which refers to the number of ways to choose \(k\) elements from \(n\) elements without regard to the order of selection.
The formula to compute a binomial coefficient is:
\[ \binom{n}{k} = \frac{n!}{k! (n-k)!} \]
Where \(!\) denotes a factorial, defined as the product of all positive integers up to that number.

• Binomial coefficients have symmetrical properties: \(\binom{n}{k} = \binom{n}{n-k}\).
• They appear in Pascal's Triangle, where each entry is the sum of the two above.
• In a binomial expansion, they represent the coefficients of each term in the series.

By calculating these coefficients, we can find the specific multipliers for corresponding terms in a binomial expansion. This makes them critical for detailed and precise algebraic manipulations.

Series Expansion

Series expansion is the representation of a function as a sum of a sequence of terms. In the case of the binomial theorem, it is the process of expressing a binomial raised to a power as an infinite or finite series.
For the function \((1 - \frac{x}{2})^4\), series expansion involves calculating each term using previously calculated binomial coefficients:
1. Start with the constant term (\(k = 0\)).2. Increase \(k\) to calculate successive terms until reaching \(k = n\).

• Each term's contribution depends on the product of the binomial coefficient and powers of \(-\frac{x}{2}\).
• Higher powers often contribute less to the sum, as seen with \(\frac{x^4}{16}\) in our example.
• This approach helps in approximating functions or solving differential equations in calculus.

Overall, series expansions enable the analysis of complex functions and facilitate approximations that simplify otherwise difficult computations.

Algebra

Algebra, a central pillar of mathematics, is the study of mathematical symbols and the rules for manipulating these symbols. It serves as a unifying thread of almost all mathematics. Through algebra, we solve equations and understand polynomial expressions, such as the expansion of binomials.
Algebraic manipulations using the binomial theorem provide insights into the nature of powers and roots:

• Allows simplification and reformation of complex expressions.
• Enables solving real-life problems involving quantities and relationships.
• Helps understand the behavior of mathematical structures under various operations.

With concepts from basic operations like addition and multiplication to high-level abstractions, algebra simplifies problem-solving across various scientific disciplines. It plays an essential role not just in expanding binomials, but in any mathematical endeavor involving manipulation of expressions.