Question

In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-2 x)^{2 / 3} $$

Short answer

The Maclaurin series is \(1 - \frac{4}{3}x + \frac{4}{9}x^2 + \cdots\).

Step-by-step solution

Identify the Series Formula

The Maclaurin series for \((1-x)^n\) is given by \(\sum_{k=0}^{\infty} \binom{n}{k} (-x)^k\). In this case, our expression is of the form \((1-2x)^{2/3}\), where \(a=2x\) and \(n=\frac{2}{3}\).

Apply the General Formula

Substitute \(a=2x\) and \(n=\frac{2}{3}\) into the series formula. The expression becomes \(\sum_{k=0}^{\infty} \binom{2/3}{k} (-(2x))^k\).

Binomial Coefficient Calculation

The binomial coefficient for real numbers is defined as \(\binom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!}\). Compute this for each \(k\) using \(n=\frac{2}{3}\). For example, \(\binom{2/3}{1} = \frac{2}{3}\), \(\binom{2/3}{2} = \frac{\frac{2}{3}\cdot(-\frac{1}{3})}{2}\).

Compute Terms

Substitute the computed coefficients into the series expression. For \(k=0\), the term is \(1\); for \(k=1\), the term is \(-\frac{4}{3}x\); for \(k=2\), the term is \(\frac{4}{9}x^2\), and the series continues.

Construct the Series

Combine the computed terms to form the Maclaurin series: \(1 - \frac{4}{3}x + \frac{4}{9}x^2 + \cdots\).

Key concepts

The Binomial Theorem

The binomial theorem is a powerful tool in mathematics, especially useful for expanding expressions of the form \((a + b)^n\). This theorem can be extended to non-integer values of \(n\), which is especially helpful in calculus when dealing with series expansions. The formula is expressed as:

• For integer \(n\), \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• For real \(n\) and in contexts like Maclaurin series, it becomes: \((1-x)^n = \sum_{k=0}^{\infty} \binom{n}{k} (-x)^k\)

When working with Maclaurin series, we often apply the theorem in the second form, which helps in expanding functions around zero. The binomial coefficient \(\binom{n}{k}\) describes the number of ways to choose \(k\) items from \(n\) items, though when \(n\) is not a positive integer, it is calculated using the formula given in calculus courses, incorporating factorials.

Series Expansion

Series expansion in calculus is the process of expressing a function as an infinite sum of terms. These terms are typically expressed in powers of a variable, like \(x\). In this context, a series offers a way to approximate functions using a list of numbers derived from the function’s derivatives at a single point.

• A Maclaurin series is a type of series expansion centered at 0. It's an application of the Taylor series but specifically about the point \(x=0\).
• For a function \(f(x)\), the Maclaurin series is \(f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots\).

What’s amazing about this is its power to help approximate functions with just a few terms. In our example, we expanded \((1-2x)^{2/3}\) by using its Maclaurin series, which gives an insightful approximation helpful for calculations in problems where a full expression is complex to deal with.

Calculus Concepts

Calculus is the mathematical study of continuous change and is at the heart of many scientific advancements. Within calculus, series and the binomial theorem allow us to delve into understanding complex behaviors of mathematical functions. Key concepts include:

• Derivatives: which measure how a function changes as its input changes. Maclaurin and Taylor series use derivatives to build their expansions.
• Limits: which help define derivatives and integrals, underlying many processes in series expansion, particularly convergence.
• Integral calculus: focuses on the accumulation of quantities and can be used with series to predict long-run behaviors.

In problems involving series expansions, differentiation and integration pave the way for simplifying limit calculations of infinite sums, making complex functions easier to handle. This is crucial in physics, engineering, and economic models where nuanced prediction of system behaviors is needed.