Question

Use the binomial series to find the Maclaurin series for the function. $$ f(x)=\sqrt{1+x^{3}} $$

Short answer

The Maclaurin series for the function \(f(x)=\sqrt{1+x^{3}}\) is \(f(x) = 1 +\frac{1}{2}x^3 -\frac{1}{8}x^6 + ...\)

Step-by-step solution

Identify the general form of the Binomial Theorem

The general form of the Binomial Theorem for any real number \(a\) is: \((1+x)^a = 1 + ax + \frac{a(a-1)x^2}{2!} + \frac{a(a-1)(a-2)x^3}{3!} + ...\). This can be applied when \(|x| < 1\), meaning that for this exercise, \(|x^3| < 1\), which implies \(|x| < 1\).

Apply the binomial theorem to the function

In our function, \(f(x)=\sqrt{1+x^{3}}\), the term inside the square root follows the structure of the binomial theorem. Thus applying the theorem where \(a = 1/2\) and replacing \(x\) with \(x^3\), we get: \(f(x) = (1 + x^3)^{1/2} = 1 + \frac{1}{2}x^3 - \frac{1}{8}(x^3)^2 + ...\).

Write the Maclaurin series for the function

After applying the binomial theorem, the function \(f(x)\) can now be expressed as a Maclaurin series: \(f(x) = 1 +\frac{1}{2}x^3 -\frac{1}{8}x^6 + ...\). Note the Maclaurin series is simply a Taylor series expansion about zero.