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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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$\sum_{n=1}^\infty\dfrac{\cos (3^nx)}{2^n}$

$   \leq$   $ \sum_{n=1}^\infty\left|\dfrac{\cos (3^nx)}{2^n}\right|$ $ $

 $\leq \sum_{n=1}^\infty\dfrac{1}{2^n} = \dfrac{1}{2}\sum_{k=0}^\infty\left(\dfrac{1}{2}\right)^k$

This is simply a geometric series with $|r| < 1$ which implies $\sum |a_n|$ Is convergent.

 We all know that if $\sum |a_n|$ converges then so is $\sum a_n$. So option D is correct.
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