How to prove $f\left( x\right) =\sum^{\infty}_{n=0}\dfrac {1}{2^{n}}\cos\left( 3^{n}x\right)$ converges uniformly on $\mathbb{R}$

Question

I understood what uniform convergence means, but I cannot even find clue how I get into proof.

Answer 1

Weierstrass M-Test to $\left|\dfrac{1}{2^{n}}\cos(3^{n}x)\right|\leq\dfrac{1}{2^{n}}$ and $\displaystyle\sum_{n}\dfrac{1}{2^{n}}<\infty$.