Showing that $\sum_{i=1}^{\infty}2^{-k}\cos(kx)$ is continuous with a continuous derivative

Here is a Real Analysis problem I have managed to develop a partial answer to and would be interested in the continuation:

Question

Consider the series $g(x)=\sum_{i=1}^{\infty}$$2^{-k}\cos(kx)$ and show that it defines a continuous series from $\mathbb{R} \rightarrow \mathbb{R}$ and has a continuous derivative.

Attempt

I know that each of the individual terms are continuous, and that, in general, the sum of continuous functions on the same interval should be continuous.

However, since this is an infinite sum, I am hesitant to make this argument. It also feels difficult to apply this to epsilon-delta definition of continuity to make any meaningful conclusion about the sum.

By this same reasoning, I am unsure about the derivative. I know that term by term differentiation is justified by the $M$-test, however, I'm sure about how to go about formulating a continuity argument here.