I believe you are talking about pointwise convergence. There are many modes of convergence, for example convergence in $L^2$, then we would have that $\|S_n f - f\|_{L^2} \to 0$ as $n \to \infty$ where $S_n f$ are the partial sums of the Fourier series of $f$ up to $n$. This is a quite easy result as the exponentials $e^{i n x}$ form a Schauder basis for $L^2$, then we can just use the inner product and use some syntax manipulation magic to obtain the result.
A much harder question however is about the pointwise convergence of the Fourier series, that is $S_n f(x) \to f(x)$ for (all?) $x$ as $n \to \infty$. This has been settled in 1966 by Carleson in Carleson's theorem. This only gives pointwise convergence almost everywhere for $L^2$ functions to the function itself. Of course, certain classes of functions even give uniform convergence to the original function.
So, your question amounts to the following. Given a function $f$, we can compute its Fourier coefficients and then write down the candidate for its Fourier series. Using different means we can prove the convergence of this series. However, proving the convergence does not imply that it actually converges to the original function! Just as you would have with Taylor series.
But, sure, we have stuff like Carleson's theorem that makes life good. :-).