PX153 - J1 - introduction

f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos (\frac{n π x}{L}) + b_{n} \sin (\frac{n π x}{L}))

for $- L \leq x \leq L$

it is not obvious that an arbitrary function, $f (x)$ , can be expressed as such in an interval, $- L \leq x \leq L$
questions:
- what functions, $f (x)$ , can be expressed as such?
- how fast/well does the sum converge?
- given a function, $f (x)$ , how can $a_{0}$ , $a_{n}$ , and $b_{n}$ be calculated?
- why is this needed?