Unlike some simpler sums, there does not appear to be any to turn your infinite sum into a small finite expression. If we could do that then that would be the answer for you.
So can we provide a convincing graphical argument that adding only a sufficient, but finite, number of terms will give a plot that cannot be distinguished from the plot of the infinite sum, if we only waited an infinite amount of time?
Compare these three plots
Plot[Sum[1/2^n Cos[4^n x], {n, 1, 10}], {x, -10, 10}]
Plot[Sum[1/2^n Cos[4^n x], {n, 11, 100}], {x, -10, 10},PlotPoints->100]
Plot[Sum[1/2^n Cos[4^n x], {n, 101, 1000}], {x, -10, 10},PlotPoints->1000]
The second plot shows summing 90 additional terms is 1000 times smaller than the first ten terms.
The third plot shows summing 900 additional terms is 10^28 times smaller than the first 100 terms.
I inserted the PlotPoints option to tell Mathematica to use even more points than normal because the sums are so small that the default number of points is too rough.