So this is the follow up thread to the one I asked before but you don't need to read the other one for this to make sense. If you want to, read PZZ's answer: link to the thread.
So I know that there exist a basis in $L^2$ which is a set of functions in the form $e^{inx}$. It turns out that this is an orthonormal basis. Also given any $f \in L^2$, there exists a sequence of complex numbers $(c_n)$ such that $ f = \sum_{n \in \mathbb{Z}}{c_ne^{inz}}$ It turns out these sequences lie in the vector space $l^2$.
What I am confused about is that in my lecture notes, I have the following derivation. Suppose $ f = \sum_{n \in \mathbb{Z}}{c_ne^{inz}}$ is the fourier series of $f :[-\pi,\pi] \rightarrow \mathbb{C}$ Then by definition: $ \begin{align} c_n &= \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx} dx \\ c_{-n} &= \overline{c_n} = \frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)e^{inx} dx \end{align} $
And so we can rewrite the fourier series as following: $\sum_{n=\infty}^{\infty}c_n e^{int} = c_0 + \sum_{n=1}^{\infty}\left( c_ne^{int}+c_{-n}e^{-int} \right)$
And this is where my problem is. If someeone could explain how the one single sum series is equal to a constant + sum of series with starting n = 1, it would be appreciated. I am basically looking for a "summary" of what fourier series are. I've googled countless pdfs and lecture but they all start of with the definition that $f(t) = c_n + \sum a_n \cos{nt} + b_n \sin{nt} $. I can derive that using my above definition (sub $c_n = a_n + ib_n$) but I'd like to know where its coming from and what fourier series actually are.