A function is define as $$ f(t) = \begin{cases} 0 & \text{ if } \pi<x\le -1 \\ x^2 & \text{ if } -1<x<1 \\ 0 & \text{ if } 1\le x < \pi \end{cases} $$ Find the number of $N$ such that $|S_n - f(x)|<\epsilon = 0.1$.
On it's solution, it begins with $$N \ge \frac{\int_{-\pi}^\pi |f'(x)|}{\pi \epsilon ^2 }$$ The paper is written in Danish, I can't understand why it begins with it. My best guess was to being with finding the coefficients (which I calculated) which is $$\epsilon = \left|\sum_{n=N+1}^\infty a_n \right| \le \sum_{n=N+1}^\infty \left | \frac{2 \left(\left(n^2-2\right) \sin (n)+2 n \cos (n)\right)}{n^3}\right|\le 2\sum_{n=N+1}^\infty \frac{n^2 + 2n-2}{n^3}$$
And approximate the series and find the value of $N$ but it diverges. Can anyone point out to textbook which formula is used or what options do I have?