Fourier series of Bernoulli polynomials

Example

Let $B_k$ be the $k$-th Bernoulli polynomial, and let $k \ge 2$ be an integer. Then for every $x \in [0, 1]$, $$B_k(x) = -\frac{k!}{(2\pi i)^k} \sum_{n \in \Z, \, n \ne 0} \frac{e^{2\pi i n x}}{n^k},$$ and the series converges absolutely and uniformly on $[0, 1]$. So the $1$-periodic function equal to $B_k$ on $[0, 1)$ has the Fourier coefficients $0$ at $n = 0$ and $-\frac{k!}{(2\pi i n)^k}$ at $n \ne 0$, for period $1$.

Check of the coefficients: $\int_0^1 B_k(x) \, dx = 0$ for $k \ge 1$, which gives the coefficient at $n = 0$. For $n \ne 0$, integration by parts with $B_k' = k B_{k-1}$ and $B_k(1) = B_k(0)$ for $k \ge 2$ gives $\int_0^1 B_k(x) e^{-2\pi i n x} \, dx = \frac{k}{2\pi i n} \int_0^1 B_{k-1}(x) e^{-2\pi i n x} \, dx$. For $k = 1$, $\int_0^1 (x - \frac{1}{2}) e^{-2\pi i n x} \, dx = -\frac{1}{2\pi i n}$, and induction gives $-\frac{k!}{(2\pi i n)^k}$. For $k \ge 2$ the coefficients are summable and the periodic extension is continuous, so the series converges uniformly to it.