Period of a trigonometric series

Question

What is the period of the function represented by the series

$a_1 cos x+a_2cos2x+a_3cos3x+...$

I guess it is $2\pi$. Am I right?

Answer 1

You are not right, not in general.

First of all, the series in question may not even converge so talking about its period would make no sense. But let's assume the series converges to a function $f(x)$

The period is at most $2\pi$, in the sense that $f(x+2\pi)=f(x)$ without a doubt. But, for example, if $a_1=a_3=a_4=\dots=0$, then $f(x+\pi)=f(x)$ as well, and the period of $f$ is $\pi$.