Mathematico | Mathematico

Show that, for appropriately small $x,$

(x + \frac{1}{x}) \sqrt[3]{1 + x} \approx \frac{1}{x} + a + b x

stating the values of $a$ and $b .$

First,

(x + \frac{1}{x}) \sqrt[3]{1 + x} = (x + \frac{1}{x}) {(1 + x)}^{\frac{1}{3}}

The $\frac{1}{x}$ will combine with the $x^{2}$ term in the bracket to make an $x$ term, so you need to find the expansion up to $x^{2}$ before multiplying out and simplifying.