Editing Periodic sequence (section)

==Periodic 0, 1 sequences==

Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions:

:<math>\sum_{k=0}^0 \cos \left(2\pi\frac{nk}{1}\right)/1 = 1,1,1,1,1,1,1,1,1, \cdots</math>

:<math>\sum_{k=0}^{1} \cos \left(2\pi\frac{nk}{2}\right)/2 = 1,0,1,0,1,0,1,0,1,0, \cdots</math>

:<math>\sum_{k=0}^{2} \cos \left(2\pi\frac{nk}{3}\right)/3 = 1, 0,0,1,0,0,1,0,0,1,0,0,1,0,0, \cdots</math>

:<math>\cdots</math>

:<math>\sum_{k=0}^{N-1} \cos \left(2\pi\frac{nk}{N}\right)/N = 1,0,0,0,\cdots,1, \cdots  
\quad \text{sequence with period } N </math>

One standard approach for proving these identities is to apply [[De Moivre's formula]] to the corresponding [[root of unity]]. Such sequences are foundational in the study of [[number theory]].