Question

Write out the first 5 terms of the Binomial series with the given \(k\) -value. $$k=1 / 2$$

Short answer

The first 5 terms are: \(1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4\).

Step-by-step solution

Understand the Binomial Series

The Binomial Series is an expansion of the form \((1 + x)^k\) where \(k\) can be any real number. It is expressed as an infinite series if \(|x| < 1\) and includes terms like \(\binom{k}{n}x^n\).

Specify the Binomial Coefficient

For the Binomial Series, the coefficient for the \(n\)-th term is \(\binom{k}{n} = \frac{k(k-1)(k-2)\cdots(k-n+1)}{n!}\). For example, the 0-th term coefficient is 1, the 1st term is \(k\), the 2nd term is \(\frac{k(k-1)}{2!}\), etc.

Substitute the Given Value of \(k\)

Here, \(k=\frac{1}{2}\). Substitute into the coefficients: - 0-th term: 1- 1st term: \(\frac{1}{2}\)- 2nd term: \(\frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}\)- 3rd term: \(\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)}{3!}\), etc.

Calculate the First Five Terms

Calculate each term using the formula from Step 3:- 0-th term: \(1\)- 1st term: \(\frac{1}{2}x\)- 2nd term: \(-\frac{1}{8}x^2\)- 3rd term: \(\frac{1}{16}x^3\)- 4th term: \(-\frac{5}{128}x^4\)

Write the Expression for the First Five Terms

Combine the calculated terms:\[1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4\] as the expression for the first five terms of the Binomial Series.

Key concepts

Binomial Coefficient

The binomial coefficient is a crucial part of understanding the binomial series expansion. It gives each term in the series its weight or significance. You may have seen it in the form \( \binom{k}{n} \), where \(k\) is usually a non-negative integer in typical combinations. However, in the binomial series, \(k\) can be any real number. This idea is at the heart of expanding a binomial expression.

• The 0-th term always has a binomial coefficient of 1 because any number raised to the power of zero is 1.
• The first term coefficient is simply \(k\).
• As you move further, the coefficients become more complex, calculated as \(\frac{k(k-1)(k-2)\ldots(k-n+1)}{n!}\).

The factorial \(n!\) is the product of all positive integers up to \(n\), which ensures that as terms expand, they do so according to combinatorial principles. Calculating these coefficients accurately provides each level of magnification of \(x\) with its proper place in the series.

Infinite Series Expansion

Infinite series expansion allows us to express functions as an infinite sum of terms. This sems to be an infinite approach but in reality serves to approximate functions to desired degrees of accuracy. In the binomial series context:

• The expression \((1+x)^k\) is expanded into an infinite series, making it versatile.
• The series is applicable only within a certain radius, specifically for \(|x| < 1\). This constraint ensures convergence of the series.
• Every term in the series adds another level of precision to the approximation.

Even though we say "infinite series," we often work with a finite number of terms for practical calculations. The more terms you include, the more precise your approximation will be. However, it is key to grasp that this infinite construct grants us powerful tools to address real-world and theoretical problems with potentially unlimited precision.

Power Series

Power series are another classical way to express a function as an infinite sum but centered around a specific point, usually at zero. This method is closely related to the binomial series:

• Each term in the binomial series can also be viewed as part of a power series where the exponent \(n\) determines the power of \(x\).
• In a power series, the "dominant" term is the one closest to the degree of the polynomial function being represented. It dictates the most immediate change in curve shape or function value.
• Power series give us the flexibility to represent and study how changes in input \(x\) precisely affect a function \(f(x)\).

By handling power series correctly, we can effectively manipulate and understand complex mathematical structures, be it the natural exponential function or simple binomials, by breaking them down into their component powers and analyzing each separately. This grants a streamlined approach to both computation and theoretical insight.