# Thread: representing function as power serries

1. ## representing function as power serries

question: $f(x)=\frac{1}{x-5}$
answer: $-\sum_{n=0}^\infty \frac{1}{5^{n+1}}x^n$ for |x|<5
work: $\frac{1}{x-5}=\frac{1}{-5} \times \frac{1}{1-x} \longrightarrow \frac{1}{-5} \sum_{n=0}^\infty x^n$

2. $\frac{1}{x-5}=-\frac{1}{5}\cdot \frac{1}{1-\frac{x}{5}}=-\frac{1}{5}\sum\limits_{n=0}^{\infty }{\left( \frac{x}{5} \right)^{n}}=-\sum\limits_{n=0}^{\infty }{\frac{x^{n}}{5^{n+1}}}.$

3. Originally Posted by superdude
question: $f(x)=\frac{1}{x-5}$
answer: $-\sum_{n=0}^\infty \frac{1}{5^{n+1}}x^n$ for |x|<5
work: $\frac{1}{x-5}={\color{red}{\frac{1}{-5} \times \frac{1}{1-x} \longrightarrow \frac{1}{-5} \sum_{n=0}^\infty x^n}}$
$f\left( x \right) = \frac{1}{{x - 5}} = - \frac{1}{{5 - x}} = - \frac{1}{5} \cdot \frac{1}{{1 - \left( {{x \mathord{\left/{\vphantom {x 5}} \right.\kern-\nulldelimiterspace} 5}} \right)}} =$

$= - \frac{1}{5}\left( {1 + \frac{x}{5} + \frac{{{x^2}}}{{25}} + \ldots + \frac{{{x^n}}}{{{5^n}}} + \ldots } \right) =$

$- \frac{1}{5}\sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{{5^n}}}} = - \sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{{5^{n + 1}}}}} ,{\text{ }}\left| x \right| < 5.$