Aperiodic Monotonic Trigonometric Divergent Series

SUPERSPECIES

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

$\displaystyle \sum_{n\geq0} (\sin n)^2 = 0 + (\sin 1)^2 + (\sin 2)^2 + (\sin 3)^2 + (\sin 4)^2 + \cdots = -\dfrac{1}{2}$ (unstable)

SPECIES

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

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

$\displaystyle \sum_{n\geq1} \sin n = \sin 1 + \sin 2 + \sin 3 + \sin 4 + \cdots = \dfrac{1}{2}\cot\dfrac{1}{2}$ (stable)

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

$\displaystyle \sum_{n\geq0} (\cos n)^2 = 1 + (\cos 1)^2 + (\cos 2)^2 + (\cos 3)^2 + (\cos 4)^2 + \cdots = 0$ (unstable)

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

$\displaystyle \sum_{n\geq1} (\sin n)^2 = (\sin 1)^2 + (\sin 2)^2 + (\sin 3)^2 + (\sin 4)^2 + \cdots = 0$ (stable)

$\displaystyle \sum_{n\geq1} (\tan n)^2 = (\tan 1)^2 + (\tan 2)^2 + (\tan 3)^2 + (\tan 4)^2 + \cdots = 0$ (stable)

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

$\displaystyle \sum_{n\geq1} (\sin n)^3 = (\sin 1)^3 + (\sin 2)^3 + (\sin 3)^3 + \cdots = \dfrac{1}{8}\left(3\cot\dfrac{1}{2}-\cot\dfrac{3}{2}\right)$ (stable)

$\displaystyle \sum_{n\geq1} (\sin n)^4 = (\sin 1)^4 + (\sin 2)^4 + (\sin 3)^4 + (\sin 4)^4 + \cdots = 0$ (stable)

$\displaystyle \sum_{n\geq1} (\sin n)^5 = (\sin 1)^5 + (\sin 2)^5 + (\sin 3)^5 + \cdots = \dfrac{1}{32}\left(10\cot\dfrac{1}{2}-5\cot\dfrac{3}{2}+\cot\dfrac{5}{2}\right)$ (stable)

$\displaystyle \sum_{n\geq1} (\sin n)^6 = (\sin 1)^6 + (\sin 2)^6 + (\sin 3)^6 + (\sin 4)^6 + \cdots = 0$ (stable)

$\displaystyle \sum_{n\geq1} (\sin n)^7 = (\sin 1)^7 + (\sin 2)^7 + (\sin 3)^7 + \cdots \\ = \dfrac{1}{128}\left(35\cot\dfrac{1}{2}-21\cot\dfrac{3}{2}+7\cot\dfrac{5}{2}-\cot\dfrac{7}{2}\right)$ (stable)

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