Answer

问题及解答

证明: $\frac{\pi}{4}=\sum\limits_{n=0}^{+\infty}\frac{\sin(2n+1)}{2n+1}$.

Posted by haifeng on 2012-07-02 17:53:08 last update 2012-07-02 17:53:08 | Edit | Answers (0)

提示: 考虑函数

\[
f(x)=
\begin{cases}
\frac{1}{2}, & 0 < x < \pi,\\
0, & x=0,\pm\pi,\\
-\frac{1}{2}, & -\pi < x < 0.\\
\end{cases}
\]

求 $f(x)$ 的 Fourier 展开, 得

\[
f(x)\sim\frac{2}{\pi}\sum_{n=0}^{+\infty}\frac{\sin(2n+1)x}{2n+1},
\]

然后取 $x=1$ 即得结论.