sinn/n和cosn/n类型的若干问题 - Eufisky - The lost book
傅里叶变换求积分函数
若干常见的傅里叶级数

sinn/n和cosn/n类型的若干问题

Eufisky posted @ 2014年4月10日 12:50 in 数学分析 , 2352 阅读
题一:求极限
\[\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos k} \right|} .\]
解:取$\displaystyle f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos \left( {k + x} \right)} \right|} $,则
\[f\left( {x + m} \right) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos \left( {k + x + m} \right)} \right|} \left( {m \in {N^*}} \right).\]
\begin{align*}f\left( x \right) &= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos \left( {k + x} \right)} \right|}  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^m {\left| {\cos \left( {k + x} \right)} \right|}  + \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = m + 1}^n {\left| {\cos \left( {k + x} \right)} \right|}  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = m + 1}^n {\left| {\cos \left( {k + x} \right)} \right|}  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{s = 1}^n {\left| {\cos \left( {s + m + x} \right)} \right|}  = f\left( {x + m} \right).\end{align*}
\[f\left( {x + 2\pi } \right) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos \left( {k + x + 2\pi } \right)} \right|}  = f\left( x \right).\]
所以
\begin{align}\label{MA1}f\left( x \right) = f\left( {x + m} \right) = f\left( {x + 2\pi } \right).\qquad(1)\end{align}
对于$\displaystyle f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos \left( {k + x} \right)} \right|} $,其中$\displaystyle \frac{1}{n}$单调且$\displaystyle\lim\limits_{n\rightarrow \infty}{\frac{1}{n}}=0,|\cos(k+x)|$在$x\in[0,1]$上一致有界,由狄利克雷判别法可知:函数$\displaystyle f(x)$一致收敛.同时(1)式说明周期可为任意整数,又可为$\displaystyle 2\pi$,那么必为常数函数,$\displaystyle f(x)\equiv C.$所以
\begin{align*}f\left( 0 \right) &= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos k} \right|}  = \int_0^1 {f\left( x \right)dx}  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\int_0^1 {\sum\limits_{k = 1}^n {\left| {\cos \left( {k + x} \right)} \right|} dx}  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\int_0^1 {\left| {\cos \left( {k + x} \right)} \right|dx} }  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\int_k^{k + 1} {\left| {\cos t} \right|dt} }  \\&= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\int_1^{n + 1} {\left| {\cos t} \right|dt}  \\&= 2\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left[ {\frac{n}{\pi }} \right] \\&= \frac{2}{\pi }\mathop {\lim }\limits_{n \to \infty } \frac{\pi }{n}\left[ {\frac{n}{\pi }} \right] = \frac{2}{\pi }\end{align*}
注:在区间$[1,n+1]$中,含有函数$\omega(t)=|\cos t|$的周期个数是$\left[ {\frac{n}{\pi }} \right]$,每个周期的积分值都是$2$,所以得到上述结果.
\[\mathop {\lim }\limits_{n \to \infty } \frac{\pi }{n}\left[ {\frac{n}{\pi }} \right] = \mathop {\lim }\limits_{n \to \infty } \frac{{\frac{n}{\pi } - \left( {\frac{n}{\pi }} \right)}}{{\frac{n}{\pi }}} = \mathop {\lim }\limits_{n \to \infty } \left( {1 - \frac{{\left( {\frac{n}{\pi }} \right)}}{{\frac{n}{\pi }}}} \right) = 1\]
其中$\left( {\frac{n}{\pi }} \right)\in(0,1).$
另解一:构建$|\cos k|,k\in\mathbb{N}^*$在$|\cos x|$上的随机分布,所以$|\cos k|$的均值就为
\[\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos k} \right|}  = E\left( {\left| {\cos k} \right|} \right) = \frac{{\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left| {\cos x} \right|dx} }}{\pi } = \frac{2}{\pi }.\]
另解二:令
\[f(x)=\int_0^x{|\cos t|dt}. \]
当$x\in[n\pi,(n+1)\pi)$时,
\[\int_0^{n\pi } {\left| {\cos t} \right|dt}  \le f\left( x \right) < \int_0^{\left( {n + 1} \right)\pi } {\left| {\cos t} \right|dt} .\]
\[\Rightarrow 2n\leq f(x)<2(n+1).\]
由此得
\[\frac{{2n}}{{\left( {n + 1} \right)\pi }} < \frac{{f\left( x \right)}}{x} < \frac{{2\left( {n + 1} \right)}}{{n\pi }}.\]
记$S_n=\sum\limits_{k = 1}^n {\left| {\cos k} \right|}$,则有
\[\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\left| {\cos k} \right|}  = \mathop {\lim }\limits_{n \to \infty } \frac{{{S_n}}}{n} = \mathop {\lim }\limits_{n \to \infty } \frac{{f\left( x \right)}}{x} = \frac{2}{\pi }.\]

题二:证明

\[\sum\limits_{n = 1}^\infty  {\frac{{\sin n}}{n}}  = \frac{{\pi  - 1}}{2}.\]
证明:由Fourier级数
\[\frac{{\pi-x}}{2} = \sum\limits_{n = 1}^\infty  {\frac{{\sin nx}}{n}} \qquad 0 < x < 2\pi\]
令$x=1$,我们立得
\[\sum\limits_{n = 1}^\infty  {\frac{{\sin n}}{n}}  = \frac{{\pi  - 1}}{2}.\]
 
题三:证明
\[\sum\limits_{n = 1}^\infty  {\frac{{\cos n}}{n}}  =  - \frac{1}{2}\ln (2 - 2\cos 1).\]
证明:由Fourier级数
\[\sum_{k=1}^\infty \frac{\cos(kx)}{k}=-\frac{1}{2}\ln(2-2\cos x)\qquad x\in\mathbb{R}. \]
令$x=1$,我们立得
\[\sum\limits_{n = 1}^\infty  {\frac{{\cos n}}{n}}  =  - \frac{1}{2}\ln (2 - 2\cos 1).\]
 
 
 
 
 

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