$$\sum_{n=1}^{\infty} cos(2^n)$$I have tried to use several convergence test without any results. Also, since $a_n=cos(2^n)$ always take positive values, $|a_n|=a_n$ and therefore if $a_n$ converges it also absolute converges, right?
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1$\cos(2)=-0.41614683\ldots$ – Eclipse Sun Oct 16 '17 at 17:43
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2Approximate $2\pi$ by a rational numbers of the form $p/2^n$ for arbitrarily large $n$. That will show you that $\cos(2^n)$ doesn't tend to zero. – Hellen Oct 16 '17 at 17:44
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1What do you know about the rotations of the circle? – Did Oct 16 '17 at 17:45
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2@Hellen I think you mean $\frac1\pi$ and the approximation must not be too bad – Hagen von Eitzen Oct 16 '17 at 17:49
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Consider $$\cos (2^{n+1}) = \cos^2 (2^n) - \sin^2(2^n).$$ – Daniel Fischer Oct 16 '17 at 18:37
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See also this. – Daniel Fischer Oct 16 '17 at 18:40