In my textbooks intro to Fourier series, it says that we can represent any periodic function with a combination of the fundamental and harmonics
fundamental = $\sin{\omega t}+\cos{\omega t}$ harmonics = $\sin{n \omega t}+\cos{n \omega t}$ so we get a summation of fundamental + harmonics
Does n have to be an integer? Does $\omega$ have to be the same for both the fundamental and the harmonics? What if it's different?
If we take $n$ to be integer and $\omega$ be the same, we don’t lose in generality; for a function with fundamental period $T$, $\omega = 1/T$ (and if the function is constant, we can take any).
Also, you’ve forgot the constant term and coefficients of all cosines and sines. And also, there is really no need to designate any of Fourier harmonics as a “fundamental” and write it alone—it may ever complicate reasoning.
