## Sum of a Geometric Series With Complex Terms

Suppose we have a geometric series with complex terms, first term
$a=1+3i$
and common ratio
$r= \frac{1+i}{3}$
.
$\| r \| = \| \frac{1+i}{3} \| = \frac{\sqrt{2}}{3} \lt 1$
so we can use the formula for the sum of a geometric series
$S= \frac{a}{1-r}$
.
\begin{aligned} S &= \frac{a}{1-r} \\ &= \frac{1+3i}{1- \frac{1+i}{3}} \\ &= \frac{1+3i}{\frac{2-i}{3}} \\ &= \frac{3+9i}{2-i} \\ &= \frac{3+9i}{2-i} \times \frac{2+i}{2+i} \\ &= \frac{6+3i+18i-9}{4+2i-2i-(-1)} \\ &= \frac{-3+21i}{5} \end{aligned}