Shifting the Center. When the function $f(x)=\frac{1}{1-x}$ is expanded as a power series with center $a=0$ we get the usual geometric series

$fa=-41c_{**}\frac{1}{1-x}={\displaystyle \underset{n=0}{\overset{}{}}}{c}_n(x41{)}^n\frac{1}{1-x}={\displaystyle \underset{n=0}{\overset{}{}}}{x}^{},-1$

Expand $fas$ a power series centered $ata=-41,$ that $is,$ find the coefficients $c_{**}in$ the series:

$\frac{1}{1-x}={\displaystyle \underset{n=0}{\overset{}{}}}{c}_n(x41{)}^n$

EXPANSION:

RADIUS $OF$ CONVERGENCE:

WORK: