I'm taking my first real analysis course and I'm trying to get a better feel about analytic functions. My understanding is that an analytic function is one which can be written as a power series. My understanding is that a power series is one of the form $\sum_n a_nx^n$. I was thinking back to Fourier series and I'm pretty sure they don't fit this form.

I'm a little curious about these trigonometric series. Are they analytic? They don't fit the form $\sum_n a_nx^n$. If they are, or can be, what are the circumstances making that so?

My big question is, what type of functions are not analytic? I know of examples such as the absolute value function and such, but why EXACTLY can they not be represented as power series? Does it have to do with smoothness? And given one that's not analytic on all of R - like, I'm guessing a Fourier series with sharp points - how can one represent a smooth piece of it in the form $\sum_n a_nx^n$?