A calculus of the absurd

15.6 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,

\begin{equation} 2 + 2e^{\frac {\pi }{10}i} + 2e^{\frac {2\pi }{10}i} + 2e^{\frac {3\pi }{10}i} + 2e^{\frac {4\pi }{10}i} \end{equation}

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

\begin{equation} \frac {2\left ( 1 - \left (e^{\frac {\pi }{10}}\right )^5 \right )} {1-e^{\frac {\pi }{10}}} \end{equation}