Question

What are the first four terms of the Binomial Series for f(x) = x / (1 + x)^1/3

Answer 1

f(x) = x(1 + x)^(-1/3)

We're interested in expanding out the (1 + x)^(-1/3), and then we can just tag on the x's by multiplication.

I'll type out the general form of the first four terms.
\[ (a + b)^n = \frac{a^n}{0!} + \frac{n a^{n-1} b}{1!} + \frac{n(n-1) a^{n-2} b^2}{2!} + \frac{n(n-1)(n-2) a^{n-3} b^3}{3!} + \cdots \\
x (a + b)^n = x\frac{a^n}{0!} + x\frac{n a^{n-1} b}{1!} + x\frac{n(n-1) a^{n-2} b^2}{2!} + x\frac{n(n-1)(n-2) a^{n-3} b^3}{3!} + \cdots \\
\text{term 1: } x*\frac{a^n}{0!} \\
\text{term 2: } x*\frac{n a^{n-1} b}{1!} \\
\text{term 3: } x*\frac{n(n-1) a^{n-2} b^2}{2!} \\
\text{term 4: } x*\frac{n(n-1)(n-2) a^{n-3} b^3}{3!} \]

Let \(n = \neg \frac{1}{3}\), \(a = 1\), and \(b = x\).