Question

Use the binomial series to find the Maclaurin series for the function. $$ f(x)=\sqrt[4]{1+x} $$

Short answer

The Maclaurin series for the function \( f(x)=\sqrt[4]{1+x}\) is \( f(x) = 1 + \frac{1}{4}x - \frac{3}{32}x^2 + \frac{5}{256}x^3 - ... \).

Step-by-step solution

Write the function in suitable form for binomial series

The function \( f(x)=\sqrt[4]{1+x}\) in the format applicable for binomial series can be written as \( (1+x)^{1/4}\).

Apply binomial series

The general form of binomial series (binomial theorem) is \( (1+x)^n = 1+nx+\frac{n(n-1)x^2}{2!}+\frac{n(n-1)(n-2)x^3}{3!}+..\). Now, for \( n=1/4 \), the function can be expanded as: \( (1+x)^{1/4} = 1 + \frac{1}{4}x - \frac{3}{32}x^2 + \frac{5}{256}x^3 - ... \).

Write down the Maclaurin series for the function

The Maclaurin series for the function is the same as the expanded binomial series. Therefore, the Maclaurin series for the function \( f(x)=\sqrt[4]{1+x}\) is \( f(x) = 1 + \frac{1}{4}x - \frac{3}{32}x^2 + \frac{5}{256}x^3 - ... \).