Answer
The power series for $f(x) $ is $\sum_{n=0}^\infty\frac{x^n}{10^{n+1}}$ and the interval of convergence is $(-10,10)$.
Work Step by Step
The function of $f(x)$ can be rewritten as $f(x)=\frac{1}{10}\cdot \frac{1}{1+x/10}$.
We know the following power series:
$\frac{1}{1-x}=\sum_{n=0}^\infty x^n$ where $|x|<1$
Replacing $x$ by $-x/10$,
$\frac{1}{1+x/10}=\sum_{n=0}^\infty (x/10)^n$ where $|-x/10|<1$
$\frac{1}{10}\cdot \frac{1}{1+x/10}=\frac{1}{10}\sum_{n=0}^\infty \frac{x^n}{10^n}$ where $|x|<10$
$f(x)=\sum_{n=0}^\infty \frac{x^n}{10^{n+1}}$ the interval of convergence is $(-10,10)$.
