Problem 16 - Essential Calculus

Question

Use the binomial series to find the Maclaurin series for the function. $$ f(x)=\frac{1}{\sqrt{1-x}} $$

Step-by-Step Solution

Answer

The Maclaurin series for the function \(f(x) = \frac{1}{\sqrt{1-x}}\) using the binomial series is \(f(x) = 1 - \frac{1}{2}x - \frac{3}{8}x^2 - \frac{5}{16}x^3 -...\).

Step 1: Understanding the Binomial Series

The Binomial series states that for any real number \( p \), \((1 + x)^p = 1 + px + \frac{p(p-1)}{2!}x^2 + \frac{p(p-1)(p-2)}{3!}x^3 +...\). The task is to express the given function in this form.

Step 2: Write the Function in Binomial Form

We want to write the function in the form of \((1 + x)^p\), so write \( f(x) = \frac{1}{\sqrt{1-x}} = (1-x)^{-1/2}\). Now, our function matches the Binomial formula with \( p = -1/2 \), and \( x \) replaced by \( -x \)

Step 3: Apply the Binomial Series

With \( p = -1/2 \) and replacing \( x \) by \( -x \), we can now expand the function using the Binomial series: \( f(x) = (1-x)^{-1/2} = 1 - \frac{1}{2}x - \frac{3}{8}x^2 - \frac{5}{16}x^3 -...\). Each subsequent term can be found by multiplying the previous term by a fraction where the numerator is decreased by 2 and the denominator is increased by 1.

Key Concepts

Binomial Series

The Binomial Series is a powerful tool used to expand expressions of the form \( (1 + x)^p \), where \( p \) is any real number. This series becomes particularly useful when dealing with functions that are not simple polynomials. The expansion takes the form:

• 1 + px + \( \frac{p(p-1)}{2!}x^2 \)
• + \( \frac{p(p-1)(p-2)}{3!}x^3 \)
• + \ldots

Each subsequent term is derived by multiplying the previous term's numerator by decrements of 1 and dividing by increasing factorial terms. This repetitive multiplication reduces the complexity of direct computation, allowing complex functions to be calculated with ease.

Function Expansion

Function expansion is a way of expressing a more complex function in a simpler form using a series of terms. This means breaking down a complicated function into a sum of simpler terms that can be more easily handled. By expressing a function like \( f(x) = (1-x)^{-1/2} \) using a series, it can be expanded into an infinite sum of terms. This transformation is particularly helpful in calculus and other fields of mathematics where approximation of values is necessary. Function expansion provides an approach that can strengthen understanding and computation of limits, derivatives, and integrals.

Power Series

A power series is an infinite series of the form \( \sum_{n=0}^\infty c_n(x-a)^n \), where \( c_n \) are coefficients and \( a \) is the center of the series. Power series are central to mathematical analysis and are used to represent functions. In our example, the function is expressed around \( x = 0 \), indicating it's a special case of power series called the Maclaurin series. The coefficients \( c_n \) in this case were computed using the binomial series expansion method. Power series serve as a way to facilitate computation and analysis of complex functions by breaking them into simpler components.

Taylor Series

The Taylor Series extends the concept of polynomial approximation by allowing a function to be expressed as an infinite sum of terms calculated from the derivatives of the function at a single point. When this point is zero, it's known as a Maclaurin series, which is what we derived for \( f(x) = \frac{1}{\sqrt{1-x}} \). The steps to move from a Taylor series to a Maclaurin series involve computing the derivatives of the function, evaluating at zero, and summing these products to form an infinite series: \(\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n\). The Taylor series not only provides an approximation of functions but also offers insight into the behavior of functions near a specific point.