In physics, we often expand functions $f(x)$ of a real variable $x$ which is piecewise continuous in the interval $(-1,1)$ along with its derivatives, in terms of Legendre polynomials $\{P_n(x)\}$ as $$f(x)=\sum\limits_{n=0}^{\infty}c_nP_n(x)$$ where $c_n$ can be obtained using $$c_n=\frac{2n+1}{2}\int\limits_{-1}^{+1}f(x)P_n(x)dx.$$
Is it always possible to do such an expansion for any $f(x)$? I'm looking for the necessary and sufficient conditions.
If you mean that the series converges in the $L^2$ sense (common in physics), then it follows from orthogonality and completeness. This Answer derives the completeness of the polynomials from the completeness of the trigonometric functions. Proof of the latter requires some functional analysis. This Answer outlines a proof ("glossing over technical details").