Given a function ${f}(x)$ , which is continuous in the region $M_1<x<M_2$. Let ${F}(x)$ be the representation of ${f}(x)$ in Fourier series , such that
${f}(x)\approx{F}(x)$ = $a_0\over2$$+\sum_{n=1}^ma_n\cos(nxw)+\sum_{n=1}^mb_n\sin(nxw)$
Then is it possible to find the minimum value for $n$ such that $t_1$<${f}(x)$-${F}(x)$<$t_2$ where $t_1,t_2$ are real numbers. To be more specific, I am interested to know, how to find the value of $n$ where the condition
$-1$<${f}(x)$-${F}(x)$<$1$ will be satisfied for a given function ${f}(x)$ .
I don't know exactly whether the expression of my doubt is mathematically correct, So in sentential form, I like to know how fast (that is at what value of n) ${F}(x)$ reaches almost near to (or within bounds of ${(-1,1)}$) for given ${f}(x)$.