Aperiodic Monotonic Trigonometric Divergent Series

SECTIONS

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

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

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

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

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

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

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

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