Question

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

Short 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-by-step solution

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.

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 \( -x \)

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.