Find the Maclaurin series for the function $f(x)=\frac{1}{\sqrt[\phantom{2}]{16-x}}$ and its radius of convergence.

Solution

We write $f(x)$ in a form where we can use the binomial series.

$\frac{1}{\sqrt[\phantom{2}]{16-x}}=\frac{1}{\sqrt[\phantom{2}]{1(1-\frac{x}{16})}}$

$=\frac{1}{(4(){)}^{-\frac{1}{2}}}$

$=\frac{1}{4}()$

Using the binomial series with $k=$ and with $x$ replaced by $-\frac{x}{16},$ we have the following.