Question

Write out the first 5 terms of the Binomial series with the given \(k\) -value. $$k=4$$

Short answer

The first five terms are: \(1 + 4x + 6x^2 + 4x^3 + x^4\).

Step-by-step solution

Understanding the Binomial Series

The Binomial series for a given power \(k\) can be represented as: \((1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n\). This represents an expansion where each term is in the form \(\binom{k}{n} x^n\), where \(\binom{k}{n}\) is the binomial coefficient for selecting \(n\) elements from \(k\).

Setting Up the Series for k = 4

Set \(k = 4\) in the binomial series formula: \((1+x)^4 = \sum_{n=0}^{\infty} \binom{4}{n} x^n\). We need to explicitly compute the first five terms where \(n = 0, 1, 2, 3, 4\).

Calculating the Terms

Calculate each term one by one:1. For \(n=0\): \(\binom{4}{0}x^0 = 1\)2. For \(n=1\): \(\binom{4}{1}x^1 = 4x\)3. For \(n=2\): \(\binom{4}{2}x^2 = 6x^2\)4. For \(n=3\): \(\binom{4}{3}x^3 = 4x^3\)5. For \(n=4\): \(\binom{4}{4}x^4 = x^4\)

Writing Out the First Five Terms

Combine the calculated terms: The first five terms of the binomial series with \(k=4\) are \(1 + 4x + 6x^2 + 4x^3 + x^4\).

Key concepts

Binomial Coefficients

Binomial coefficients play a crucial role in the expansion of binomial expressions. They are written as \( \binom{k}{n} \) and represent the number of ways to choose \( n \) elements from a set of \( k \) elements without considering the order. In the context of a binomial series, these coefficients help determine the multiple of each term. For example, in the exercise provided, when expanding \((1+x)^4\), the coefficients \(1, 4, 6, 4,\) and \(1\) corresponding to the terms \(x^0, x^1, x^2, x^3,\) and \(x^4\) can be calculated using binomial coefficients: - \( \binom{4}{0} = 1 \) - \( \binom{4}{1} = 4 \) - \( \binom{4}{2} = 6 \) - \( \binom{4}{3} = 4 \) - \( \binom{4}{4} = 1 \) Each of these values corresponds to a term in the series, influencing the term's magnitude due to its calculation from the binomial coefficient formula, \( \binom{k}{n} = \frac{k!}{n!(k-n)!} \).

Series Expansion

A series expansion is a way of representing a function as a sum of simpler terms. This process takes a complex expression and breaks it down into a series of terms with gradually increasing powers of a variable, usually \(x\). For example, in the binomial series, given as \( (1+x)^k \), the series is expressed as the sum: \( \sum_{n=0}^{\infty} \binom{k}{n} x^n \). Each term in this series is a component of the polynomial expansion.In practical situations, you won't always need to compute all terms up to infinity. Often, identifying just a few terms will provide a sufficient approximation. In our example, with \( k = 4 \), the first five terms of the expansion were required. These provide an excellent look into how the series evolves, starting from a simple constant term (when \(n=0\)) and adding complexity with terms like \(4x, 6x^2\), etc.

Combinatorics

Combinatorics is a branch of mathematics concerning the study of counting, arrangement, and optimization. The binomial coefficient that was previously discussed is a fundamental tool in combinatorics. Its ability to represent a count of combinations (ways of selecting elements from a set) underpins its application to binomial series. In our series expansion, calculating the binomial coefficients involved choosing \(n\) elements out of \(k\). This is directly connected to the concept of combinations, wherein the order of selection does not matter. This attribute of combinatorics helps simplify complex algebraic expressions such as polynomial expansions, as it reduces the number of variations to consider.

Polynomial Expansion

Polynomial expansion refers to expressing a power of a binomial as a sum of terms involving powers of its components. Each term in the polynomial is a product of a binomial coefficient and a power of \(x\) from the expansion formula, \((1+x)^k\). For example, the polynomial expansion of \((1+x)^4\) is written as \(1 + 4x + 6x^2 + 4x^3 + x^4\). Here:- The polynomial expands from a single term, reaching up to \(k\) terms. - Each term \(\binom{4}{n}x^n\) builds on the previous one by increasing the power of \(x\), and modifying the coefficient according to the binomial formula.This systematic build-up makes polynomial expansions like these particularly useful, as they simplify functions to a form that's easy to interpret and compute for small values of \(x\). It also forms a foundational concept for deeper exploration in calculus and algebraic studies.