## Calculus 10th Edition

$x \lt -2$ or $x>2$ $f(x)=\displaystyle \frac{x}{x-2}, \quad x>2$ or $x \lt -2$
We aim for terms of Th.9.6 (Convergence of a Geometric Series). A geometric series with ratio $r$ diverges when $|r| \geq 1$. If $0 \lt |r| \lt 1$, then the series converges to the sum $\displaystyle \sum_{n=0}^{\infty}ar^{n}=\frac{a}{1-r},\quad 0 \lt |r| \lt 1$. ------------ $\displaystyle \sum_{n=0}^{\infty}(\frac{2}{x})^{n}$ The series is geometric, r=$\displaystyle \frac{2}{x}$, converges for $|\displaystyle \frac{2}{x}| \lt 1 \Rightarrow |x|>2 \Rightarrow x \lt -2$ or $x>2$ $f(x)=\displaystyle \sum_{n=0}^{\infty}(\frac{2}{x})^{n}=\frac{1}{1-\frac{2}{x}}=\frac{1}{\frac{x-2}{x}}$ $=\displaystyle \frac{x}{x-2}, \quad x>2$ or $x \lt -2$