@AMDG What I can understand is that you seem to be talking about something like uniform convergence for a sequence of functions. Anyway, the term thing that you were going about doesn't make sense in this constant, because to approach closer you are going closer to it by increasing the no. of terms you consider in your function.
Also not that such a $g$ is not unique. So I don't know what you mean by what such a $g$ is for a $f$. You can approximate a function by a lot of different kinds of sequence of functions (depending upon wether you want pointwise convergence or uniform)