Going from Fourier sum to Fourier integral - confusion on intermediate step How does it solve the $A_0$ term?
Can one prove this with an example? I don't know how to prove it to myself with an example
In my lecture notes, my lecturer is trying to justify $C_n$ as weighting the sine and cosine terms in the Fourier integral differently despite how it looks explicitly in the Fourier infinite sum, that is, in:


$$\sum_{n=-\infty}^{\infty} C_n \ e^{ik_nx} \ $$
I have two main confusions with this:
1) How does this rewriting with $a_n$ and $b_n$ resolve the $A_0$ term?
2) I don't see how the substitution of $A_n$ and $B_n$ for $a_n$ and $b_n$ is valid, even given the changed summation ranges. I also don't know how I'd prove it with an example. How can one do this?

The "trick" is in the sum from $-\infty$ to $\infty$ where $|n|$ is counted once for $|n|=0$ , and twice for $|n| \ge 1$.
