I am watching this video on complex Fourier Series where the instructor states the formula as:
$$
f(x) = C_0 + \sum_{-\infty}^{\infty}C_ne^{inx}
$$
where as the notes on the same topic by Cambridge Uni state the formula as:
$$
f(x) = \sum_{-\infty}^{\infty}C_ne^{inx}
$$
Which is the right formula?
Both formulas say the same thing, since the first one should be written more precisely as $$f(x) = C_0 + \sum_{n\in\mathbb Z\setminus\{0\}} C_ne^{inx}$$ With complex Fourier series, there is little reason to separate the $0$th term from the rest of the sum. (One situation when it's done is when you want to write down $\int f(x)\,dx$.)
With trigonometric series it's very common to separate the $0$th term: $$f(x)=A_0+\sum_{n=1}^\infty (A_n\cos nx+B_n\sin nx)$$ because including $B_0\sin 0x$ feels silly.