Question

Find a power series representation for the function and determine the interval of convergence. (give your power series representation centered at x = 0.) f(x)=1/(8+x)

Answer 1

$f(x)=\dfrac1{8+x}=\dfrac18\dfrac1{1-\left(-\frac x8\right)}$

Recall that

$\dfrac1{1-x}=\displaystyle\sum_{n\ge0}x^n$

for $|x|$, so we can write $f(x)$ as the series

$f(x)=\displaystyle\frac18\sum_{n\ge0}\left(-\frac x8\right)^n$