Studying the convergence and absolute convergence of $\sum_{n=1}^{\infty} cos(2^n)$.

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$$\sum_{n=1}^{\infty} cos(2^n)$$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?