Substituting the <Fourier coefficient>s into the <Fourier partial sum> and summing the finite geometric series gives
$$
s_n(f,x)=\frac1{2\pi}\int_{\mathbb T}f(t)\sum_{k=-n}^ne^{ik(x-t)}dt
=\frac1\pi\int_{\mathbb T}D_n(x-t)f(t)dt,
$$
where
$$
\boxed{D_n(u)=\frac{\sin((n+\tfrac12)u)}{2\sin(u/2)}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-318/2/a/solution