I've been studying solving homogeneous heat equation with Dirichlet boundary conditions and on the notes I encounter Fourier analysis, which I haven't figure out yet. (The notes are here)Notes
One thing I'm not sure about is that, if any function (such as f(x) = x) that satisfies the hypothesis can be approximated with an infinite sine series (sin(mπx)), for the Fourier expansion of f(x) = x with sin(mπx), when x = 1, any sin(mπx) will rigorously equal to 0, how can the summation of infinite 0 be 1?
This keeps annoying me to doubt whether I fully understand the theorem on the notes.