Show that $\displaystyle \sum_{n=0}^{\infty} \frac{(z+i)^n}{2^n}$ converges for all values of z in the disk D(-1)={z:abs(z+i)<2} and diverges if abs(z+i)>2.
abs() means absolute value.
This is a power series centered at $\displaystyle -i$
Use the generalized ratio test for power series.
That is,
$\displaystyle \lim_{n\to\infty}\frac{(1/2)^{n+1}}{(1/2)^n}=1/2$
That means the series converges absolutely when $\displaystyle |z+i|$ is strictly less then reciprocal of 1/2=2.
Thus,
$\displaystyle |z+i|<2$---> converges (absolutely)
$\displaystyle |z+i|>2$---> diverges