14.6 Helpful things to know about the exponential form ‣ Chapter 14 Complex numbers ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 14 Complex numbers ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

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)