关于反常积分收敛的专题讨论

$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，若$lim limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} fleft( x ight)$存在，则$lim limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} fleft( x ight) = 0$

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$f命题：$设$fleft( x ight) in {C^1}left[ {a, + infty } ight)$，若$int_a^{ + infty } {fleft( x ight)dx} ，int_a^{ + infty } {f'left( x ight)dx}$均收敛，则$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} fleft( x ight) = 0$

1

$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且${fleft( x ight)}$在$left[ {a,{ m{ + }}infty } ight)$单调，则$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} xfleft( x ight) = 0$，进而$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} fleft( x ight) = 0$

1

$f命题：$ 设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且可微函数${fleft( x ight)}$在$left[ {a,{ m{ + }}infty } ight)$单调递减，则$int_a^{ + infty } {xf'left( x ight)dx} $收敛

1

$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且$frac{{fleft( x ight)}}{x}$在${left[ {a, + infty } ight)}$上单调递减，则$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} xfleft( x ight) = 0$

1

$f命题：$设$fleft( x ight)$单调且$lim limits_{x o egin{array}{*{20}{c}}
{{0^ + }}
end{array}} fleft( x ight) = + infty $，若$int_0^1 {fleft( x ight)dx} $收敛，则$lim limits_{x o egin{array}{*{20}{c}}
{{0^ + }}
end{array}} xfleft( x ight) = 0$

1

$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且$xfleft( x ight)$在${left[ {a, + infty } ight)}$上单调递减，则$lim limits_{x oegin{array}{*{20}{c}} { + infty }end{array}} xfleft( x ight)ln x = 0$

1

$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且$fleft( x ight)$在${left[ {a, + infty } ight)}$上一致连续，则$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} fleft( x ight) = 0$

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$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且$fleft( x ight)$在${left[ {a, + infty } ight)}$上可导且导函数有界，则$lim limits_{x o egin{array}{*{20}{c}}
{ + infty }
end{array}} fleft( x ight) = 0$

$f命题：$设$int_a^{ + infty } {fleft( x ight)dx} $绝对收敛，且$fleft( x ight)$在${left[ {a, + infty } ight)}$上可导且导函数有界，则$lim limits_{x o egin{array}{*{20}{c}}
{ + infty }
end{array}} fleft( x ight) = 0$

1

$f命题：$设$fleft( x ight)$在${left[ {a, + infty } ight)}$上可导且导函数有界，若$ int_a^{ + infty } {{f^2}left( x ight)dx} < + infty $，则$lim limits_{x o egin{array}{*{20}{c}}
{ + infty }
end{array}} fleft( x ight) = 0$

$f命题：$设$p ge 1,fleft( x ight) in {C^1}left( { - infty , + infty } ight)$，且[int_{ - infty }^{ + infty } {{{left| {fleft( x ight)} ight|}^p}dx} < + infty ,int_{ - infty }^{ + infty } {{{left| {f'left( x ight)} ight|}^p}dx} < + infty ]
证明：$lim limits_{x o egin{array}{*{20}{c}}infty end{array}} fleft( x ight) = 0$，且$${left| {fleft( x ight)} ight|^p} le frac{{p - 1}}{2}int_{ - infty }^{ + infty } {{{left| {fleft( t ight)} ight|}^p}dt} + frac{1}{2}int_{ - infty }^{ + infty } {{{left| {f'left( t ight)} ight|}^p}dt}$$

1

$f命题：$设$fleft( x ight) in Cleft[ {a, + infty } ight)$，且$int_a^{ + infty } {fleft( x ight)dx} $收敛，则存在数列$left{ {{x_n}} ight} subset left[ {a, + infty } ight)$，使得$lim limits_{n oinfty } {x_n} = + infty ，lim limits_{n o infty } fleft( {{x_n}} ight) = 0$

1

$f命题：$设$int_a^{{ m{ + }}infty } {fleft( x ight)dx} $绝对收敛，且$lim limits_{x o egin{array}{*{20}{c}}{{ m{ + }}infty }end{array}} fleft( x ight) = 0$，则$int_a^{{ m{ + }}infty } {{f^2}left( x ight)dx} $收敛 

1

$f命题：$设$fleft( x ight)$在$left[ {0, + infty } ight)$上可微，$f'left( x ight)$在$left[ {0, + infty } ight)$上单调递增且无上界，则$int_0^{ + infty } {frac{1}{{1 + {f^2}left( x ight)}}dx} $收敛

1

$f命题：$设$fleft( x ight) in {C^1}left[ {0, + infty } ight),fleft( 0 ight) > 0,f'left( x ight) geqslant 0,int_0^{ + infty } {frac{1}{{fleft( x ight) + f'left( x ight)}}dx}  <  + infty $，证明：$int_0^{ + infty } {frac{1}{{fleft( x ight)}}dx}  <  + infty $

1

$f命题：$设正值函数$fleft( x ight)$在$left[ {1, + infty } ight)$上二阶连续可微，且$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} f''left( x ight) = + infty $，则$int_1^{ + infty } {frac{1}{{fleft( x ight)}}dx} $收敛

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$f命题：$

附录

$f(Dirichlet判别法)$设$int_a^A {fleft( x ight)dx} $在$left[ {a, + infty } ight)$上有界，且$g(x)$在$left[ {a, + infty } ight)$上单调趋于$0$，则$int_a^{ + infty } {fleft( x ight)gleft( x ight)dx} $收敛

1

$f(Abel判别法)$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，且$g(x)$在$left[ {a, + infty } ight)$上单调有界，则$int_a^{ + infty } {fleft( x ight)gleft( x ight)dx} $收敛

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 关于反常积分收敛专题的练习题