关于函数项级数一致收敛的专题讨论

$\bf命题：$设$f(x)$在[-1,1]上有连续的导函数，且$f(0)=0$

   $(1)$证明：$\sum\limits_{n = 1}^\infty  {\frac{1}{n}f\left( {\frac{x}{n}} \right)} $在[-1,1]上一致收敛

   $(2)$设$S\left( x \right) = \sum\limits_{n = 1}^\infty  {\frac{1}{n}f\left( {\frac{x}{n}} \right)} $，证明：$S\left( x \right)$在[-1,1]上连续可导

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$\bf命题：$$\bf(14武大七)$设函数项级数$\sum\limits_{n = 1}^\infty  {\frac{{{n^{n + 2}}}}{{{{\left( {1 + nx} \right)}^n}}}} $，则

   $(1)$级数在$\left( {1, + \infty } \right)$上收敛

   $(2)$级数在$\left( {1, + \infty } \right)$上非一致收敛，但在$\left( {1, + \infty } \right)$上连续

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$\bf命题：$$\bf(04大连理工)$设${u_n}\left( x \right)\left( {n = 1,2, \cdots } \right)$在$[a,b]$上可微，$\sum\limits_{n = 1}^\infty  {{u_n}\left( x \right)} $在${x_0} \in \left[ {a,b} \right]$处收敛，$\sum\limits_{n = 1}^\infty  {{u_n}^\prime \left( x \right)} $在$[a,b]$上一致收敛，证明：$\sum\limits_{n = 1}^\infty  {{u_n}\left( x \right)} $在$[a,b]$上一致收敛

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$\bf命题：$当$|x|<r$时，$\sum\limits_{n = 0}^\infty  {{a_n}{x^n}} $收敛于$f(x)$，证明：当${\frac{{{a_n}}}{{n + 1}}{r^{n + 1}}}$收敛时，$\int_0^r {f\left( x \right)dx}  = \sum\limits_{n = 0}^\infty  {\frac{{{a_n}}}{{n + 1}}{r^{n + 1}}} $

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$\bf命题：$$\bf(09武大九)$设${u_n}\left( x \right) = \frac{1}{{{n^3}}}\ln \left( {1 + {n^3}x} \right),n = 1,2, \cdots $，记$S\left( x \right) = \sum\limits_{n = 1}^\infty  {{u_n}\left( x \right)} $

(1)证明：$\sum\limits_{n = 1}^\infty  {{u_n}\left( x \right)} $在$[0,b]$上一致收敛，而在$\left( {0, + \infty } \right)$上非一致收敛

(2)讨论$S(x)$的可微性

$\bf命题：$

附录

$\bf命题：$$（\bf{Bendixon判别法}）$设$\sum\limits_{n = 1}^\infty {{u_n}\left( x \right)} $为$\left[ {a,b} \right]$上的可微函数项级数，且$\sum\limits_{n = 1}^\infty {{u_n}^\prime \left( x \right)} $的部分和函数列在$\left[ {a,b} \right]$上一致有界

证明：如果$\sum\limits_{n = 1}^\infty {{u_n}\left( x \right)} $在$\left[ {a,b} \right]$上收敛，则必在$\left[ {a,b} \right]$上一致收敛

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$\bf命题：$$（\bf{Dini定理}）$设函数项级数$\sum\limits_{n = 0}^\infty {{u_n}\left( x \right)} $的每一项及其和函数均在$\left[ {a,b} \right]$上连续，且对每个$x \in \left[ {a,b} \right]$，

有$\sum\limits_{n = 0}^\infty {{u_n}\left( x \right)} $是正项级数或负项级数，则$\sum\limits_{n = 0}^\infty {{u_n}\left( x \right)} $在$x \in \left[ {a,b}\right]$上一致收敛

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