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If $x$ is of the form $\dfrac{a\pi}{2^b}$ (where $a$ and $b$ are integers) then $\cos^n (2^n x)$ will take the value 1 infinitely often. That deals with showing that the series diverges on a dense set.Show that \(\displaystyle \sum_{n=1}^\infty \cos^n (2^n x)\) converges for a.e. x, but diverges on a dense set of x’s .