How can we show that$$\max_{x \in [0,1]} |B_k(x)| = k! 2^{1-k} \pi^{-k}(1-3^{-k} + O(4^{-k})),$$where $B_k(x)$ is the $k$-th Bernoulli polynomial and $k$ is an odd integer $\geq 3$?
From the Fourier expansion of $B_k(\{x\})$, we have$$B_k(\{x\}) = -k! (2i)^{1-k} \pi^{-k} \sum_{m > 0} m^{-k} \sin(2 \pi m x).$$
It seems that the only step needed is an asymptotic expansion of the sum above at the point $x_0$ where $|B_k(x)|$ reaches its maximum, which should yield a result of the form $1 - 3^{-k} + O(4^{-k})$. However, I am struggling to see how to achieve this.