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The series $\sum_{n=1}^{\infty}\frac{\cos (3^{n}x)}{2^{n}}$

1. Diverges, for all rational $x \in \mathbb{R}$
2. Diverges, for some irrational $x \in \mathbb{R}$
3. Converges, for some but not all $x \in \mathbb{R}$
4. Converges, for all $x \in \mathbb{R}$

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