2020 Putnam A6

For a positive integer $N$, let $f_N$\footnote{Corrected from $F_N$ in the source.} be the function defined by

$$

f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x).

$$

Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$.

对于正整数 $N$，令 $f_N$\footnote{Corrected from $F_N$ in the source.} 为由下式定义的函数

$$

f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x)。

$$

确定最小常数 $M$，使得对于所有 $N$ 和所有实数 $x$ 来说 $f_N(x) \leq M$。