Aperiodic Alternate Trigonometric Divergent Series

SPECIES

Alternate trigonometric divergent series of cos (starting from 0)

$\displaystyle \sum_{n\geq0} (-1)^n \cos(n) = \cos(0)-\cos(1)+\cos(2)-\cos(3)\pm\cdots = \dfrac{1}{2}$ (stable)

Alternate trigonometric divergent series of cos (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \cos(n) = \cos(1)-\cos(2)+\cos(3)-\cos(4)\pm\cdots = \dfrac{1}{2}$ (stable)

Alternate trigonometric divergent series of sin (starting from 0)

$\displaystyle \sum_{n\geq0} (-1)^n \sin(n) = \sin(0)-\sin(1)+\sin(2)-\sin(3)\pm\cdots = -\dfrac{1}{2}\tan\dfrac{1}{2}$ (stable)

Alternate trigonometric divergent series of sin (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \sin(n) = \sin(1)+\sin(2)+\sin(3)+\cdots = \dfrac{1}{2}\tan\dfrac{1}{2}$ (stable)