# 14.6 Helpful things to know about the exponential form

Complex numbers sometimes form geometric series (particularly when trigonometric functions are translated into exponential form). For example,

$2+2e^{\frac{\pi}{10}i}+2e^{\frac{2\pi}{10}i}+2e^{\frac{3\pi}{10}i}+2e^{\frac{4%
\pi}{10}i}$
(14.55)

forms a geometric series with $a=2$ and $r=2e^{\frac{\pi}{10}i}$. This means that the sum (as shown in the "sequences and series" section of these notes) is equal to

$\frac{2\left(1-\left(e^{\frac{\pi}{10}}\right)^{5}\right)}{1-e^{\frac{\pi}{10}}}$
(14.56)