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+2eπ10i+2e2π10i+2e3π10i+2e4π10i2+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=2a=2 and r=2eπ10ir=2e^{\frac{\pi}{10}i}. This means that the sum (as shown in the "sequences and series" section of these notes) is equal to

2(1(eπ10)5)1eπ10\frac{2\left(1-\left(e^{\frac{\pi}{10}}\right)^{5}\right)}{1-e^{\frac{\pi}{10}}} (14.56)