📝 题目
例 1 将函数 $f\left( x\right) = x\left( {-\pi \leq x \leq \pi }\right)$ 展开为傅氏级数.
💡 答案与解析
解 因为 $f\left( x\right)$ 是奇函数,所以 ${a}_{n} = 0\left( {n \geq 0}\right)$ ,
$$ {b}_{n} = \frac{2}{\pi }{\int }_{0}^{\pi }x\sin {nx}\mathrm{\;d}x = - \frac{2}{n\pi }{\left\lbrack x\cos nx\right\rbrack }_{0}^{\pi } + \frac{2}{nx}{\int }_{0}^{\pi }\cos {nx}\mathrm{\;d}x $$
$$ = {\left( -1\right) }^{n - 1}\frac{2}{n}. $$
因为 $f\left( x\right)$ 逐段单调,所以
$$ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}\frac{2}{n}\sin {nx} = \left\{ \begin{array}{ll} x, & - \pi < x < \pi , \\ 0, & x = - \pi ,\pi . \end{array}\right. $$