Here are different ways to write a Fourier series:
In Wikipedia:
$$f(x) =\sum_{n=-N}^N c_n\cdot \mathrm e^{i\frac{2\pi nx}{P}}\tag 1$$
$$f(x) =\sum_{n=-\infty}^\infty c_n\cdot \mathrm e^{i\; nx} \tag 2$$
Or on a lecture by Prof. Brad Osgood:
$$f(x) =\sum_{n=-N}^N c_n\cdot \mathrm e^{i\; 2\pi n x} \tag 3$$
I see that the difference between (1) and (3) is the premise that $P=1$ in the Stanford lecture.
However, I want to ask for some insight about when $2\pi$ is needed - one obvious difference is the sum limits with $-N$ to $N$ calling for $2 \pi$ versus $-\infty$ to $\infty$ in (2) - the only formula without $2\pi.$ But what is the reason for this different formulation?
Just for the sake of closing the question (thanks to @Shinaolord for the comments), the equation is predictably the same in the three instances:
$$f(x) =\sum_{n=-N}^N c_n\cdot \mathrm e^{i\;2\pi\;\frac{ n}{P}\;x}$$
This would be the finite or partial-sum FS of period $P$ (equivalent to the discrete Fourier transform (DFT)), as opposed to the infinite sum or Fourier series representation of $f(x)$ in equation (2) (equivalent to the discrete-time Fourier transform (DTFT)).
In addition, equation (2) uses a $P=2\pi.$ In the case of equation (3), $P=1.$