If I expand a function $f(x)$ into an infinite sum of polynomials, according to any given set of rules, i.e. Taylor series, Legendre polynomials, or other types of polynomial expansion, if the function is continous on this compact interval and can be expanded into these various types of polynomial expansions, are the coefficients of these expansions term by term equivalent?If they're not, then the difference of two different expansions wouldn't be zero, and thus they would be expanding different functions.
What's the error here?