Series approximations of functions can extend beyond Taylor/McLaurin and Fourier series, with the possibility of using Gaussian functions like Aexp(-bx^2) for expansions. The key requirement is that the functions in the series must form a basis for the function space. This principle is rooted in functional analysis, where the basis functions, such as sine and cosine in Fourier expansions, allow for accurate function representation. The discussion highlights the flexibility in choosing series for function approximation, emphasizing the importance of basis functions. Various series can be explored for effective function representation.