Range of convergence

For what values of x \in \mathbb{R} does the series

    \[\mathcal{S} = \sum_{n=1}^{\infty} \cos (2^n x)\]

converge?

For none. Since for the sequence a_n=\cos 2^n x it holds that a_{n+1} = 2 a_n^2 -1 we deduce that the potential limits are 1 or -\frac{1}{2}. Hence, the sequence cannot tend to zero and the conclusion follows.

Note: One interesting question is the following. For what values of x \in \mathbb{R} does the series

    \[\mathcal{S} = \sum_{n=1}^{\infty} \sin (2^n x)\]

converge? The answer may be found at this link.

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