關於反常積分收斂的專題討論

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$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，若$\lim \limits_{x \to \begin{array}{*{20}{c}} {{\rm{ + }}\infty } \end{array}} f\left( x \right)$存在，則$\lim \limits_{x \to \begin{array}{*{20}{c}} {{\rm{ + }}\infty } \end{array}} f\left( x \right) = 0$

方法一

$\bf命題：$設$f\left( x \right) \in {C^1}\left[ {a, + \infty } \right)$，若$\int_a^{ + \infty } {f\left( x \right)dx} ，\int_a^{ + \infty } {f‘\left( x \right)dx}$均收斂，則$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f\left( x \right) = 0$

方法一

$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且${f\left( x \right)}$在$\left[ {a,{\rm{ + }}\infty } \right)$單調，則$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} xf\left( x \right) = 0$，進而$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f\left( x \right) = 0$

方法一

$\bf命題：$ 設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且可微函數${f\left( x \right)}$在$\left[ {a,{\rm{ + }}\infty } \right)$單調遞減，則$\int_a^{ + \infty } {xf‘\left( x \right)dx} $收斂

方法一

$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且$\frac{{f\left( x \right)}}{x}$在${\left[ {a, + \infty } \right)}$上單調遞減，則$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} xf\left( x \right) = 0$

方法一

$\bf命題：$設$f\left( x \right)$單調且$\lim \limits_{x \to \begin{array}{*{20}{c}}
{{0^ + }}
\end{array}} f\left( x \right) = + \infty $，若$\int_0^1 {f\left( x \right)dx} $收斂，則$\lim \limits_{x \to \begin{array}{*{20}{c}}
{{0^ + }}
\end{array}} xf\left( x \right) = 0$

方法一

$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且$xf\left( x \right)$在${\left[ {a, + \infty } \right)}$上單調遞減，則$\lim \limits_{x \to\begin{array}{*{20}{c}} { + \infty }\end{array}} xf\left( x \right)\ln x = 0$

方法一

$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上一致連續，則$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f\left( x \right) = 0$

方法一   方法二

$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上可導且導函數有界，則$\lim \limits_{x \to \begin{array}{*{20}{c}}
{ + \infty }
\end{array}} f\left( x \right) = 0$

$\bf命題：$設$\int_a^{ + \infty } {f\left( x \right)dx} $絕對收斂，且$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上可導且導函數有界，則$\lim \limits_{x \to \begin{array}{*{20}{c}}
{ + \infty }
\end{array}} f\left( x \right) = 0$

方法一

$\bf命題：$設$f\left( x \right)$在${\left[ {a, + \infty } \right)}$上可導且導函數有界，若$ \int_a^{ + \infty } {{f^2}\left( x \right)dx} < + \infty $，則$\lim \limits_{x \to \begin{array}{*{20}{c}}
{ + \infty }
\end{array}} f\left( x \right) = 0$

$\bf命題：$設$p \ge 1,f\left( x \right) \in {C^1}\left( { - \infty , + \infty } \right)$，且\[\int_{ - \infty }^{ + \infty } {{{\left| {f\left( x \right)} \right|}^p}dx} < + \infty ,\int_{ - \infty }^{ + \infty } {{{\left| {f‘\left( x \right)} \right|}^p}dx} < + \infty \]
證明：$\lim \limits_{x \to \begin{array}{*{20}{c}}\infty \end{array}} f\left( x \right) = 0$，且$${\left| {f\left( x \right)} \right|^p} \le \frac{{p - 1}}{2}\int_{ - \infty }^{ + \infty } {{{\left| {f\left( t \right)} \right|}^p}dt} + \frac{1}{2}\int_{ - \infty }^{ + \infty } {{{\left| {f‘\left( t \right)} \right|}^p}dt}$$

方法一

$\bf命題：$設$f\left( x \right) \in C\left[ {a, + \infty } \right)$，且$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，則存在數列$\left\{ {{x_n}} \right\} \subset \left[ {a, + \infty } \right)$，使得$\lim \limits_{n \to\infty } {x_n} = + \infty ，\lim \limits_{n \to \infty } f\left( {{x_n}} \right) = 0$

方法一

$\bf命題：$設$\int_a^{{\rm{ + }}\infty } {f\left( x \right)dx} $絕對收斂，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{{\rm{ + }}\infty }\end{array}} f\left( x \right) = 0$，則$\int_a^{{\rm{ + }}\infty } {{f^2}\left( x \right)dx} $收斂 

方法一

$\bf命題：$設$f\left( x \right)$在$\left[ {0, + \infty } \right)$上可微，$f‘\left( x \right)$在$\left[ {0, + \infty } \right)$上單調遞增且無上界，則$\int_0^{ + \infty } {\frac{1}{{1 + {f^2}\left( x \right)}}dx} $收斂

方法一

$\bf命題：$設正值函數$f\left( x \right)$在$\left[ {1, + \infty } \right)$上二階連續可微，且$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} f‘‘\left( x \right) = + \infty $，則$\int_1^{ + \infty } {\frac{1}{{f\left( x \right)}}dx} $收斂

方法一

$\bf命題：$

附錄

$\bf(Dirichlet判別法)$設$\int_a^A {f\left( x \right)dx} $在$\left[ {a, + \infty } \right)$上有界，且$g(x)$在$\left[ {a, + \infty } \right)$上單調趨於$0$，則$\int_a^{ + \infty } {f\left( x \right)g\left( x \right)dx} $收斂

方法一

$\bf(Abel判別法)$設$\int_a^{ + \infty } {f\left( x \right)dx} $收斂，且$g(x)$在$\left[ {a, + \infty } \right)$上單調有界，則$\int_a^{ + \infty } {f\left( x \right)g\left( x \right)dx} $收斂

方法一

 

 

 

 

 

 關於反常積分收斂專題的練習題

關於反常積分收斂的專題討論