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拉比判别法

无穷级数 ζ ( s ) = ∑ k = 1 ∞ 1 k s {\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s)))) 无穷级数 审敛法 项测试 · 比较审敛法 · 极限比较审敛法 ·根值审敛法 · 比值审敛法 · 柯西判别法 · 柯西并项判别法 · 拉比判别法 · 高斯判别法 · 积分判别法 · 魏尔施特拉斯判别法 · 贝特朗判别法 · 狄利克雷判别法 · 阿贝尔判别法 · 库默尔判别法 · 斯托尔兹—切萨罗定理 · 迪尼判别法 级数 调和级数 · 调和级数 · 幂级数 · 泰勒级数 · 傅里叶级数 .mw-parser-output .hlist ul,.mw-parser-output .hlist ol{padding-left:0}.mw-parser-output .hlist li,.mw-parser-output .hlist dd,.mw-parser-output .hlist dt{margin:0;display:inline}.mw-parser-output .hlist dt:after,.mw-parser-output .hlist dd:after,.mw-parser-output .hlist li:after{white-space:normal}.mw-parser-output .hlist dt:after{content:" :"}.mw-parser-output .hlist dd:after,.mw-parser-output .hlist li:after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd:after,.mw-parser-output .hlist-pipe li:after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd:after,.mw-parser-output .hlist-hyphen li:after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd:after,.mw-parser-output .hlist-comma li:after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child:after,.mw-parser-output .hlist dt:last-child:after,.mw-parser-output .hlist li:last-child:after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li:before{content:" "counter(listitem)" ";white-space:nowrap}.mw-parser-output .hlist dd ol>li:first-child:before,.mw-parser-output .hlist dt ol>li:first-child:before,.mw-parser-output .hlist li ol>li:first-child:before{content:" ("counter(listitem)" "}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li:before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child:before,.mw-parser-output .hlist dt ol>li:first-child:before,.mw-parser-output .hlist li ol>li:first-child:before{content:" ("counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li:after{content:","}.mw-parser-output .sslist li:after{content:";"}.mw-parser-output .cslist li:last-child:after,.mw-parser-output .sslist li:last-child:after{content:none}.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}

拉比判别法(英语:Raabe's Test)是判断一个级数收敛的方法。在判断比几何级数收敛得慢的级数时,比柯西判别法达朗贝尔判别法更有效。[1]

定理

对任意级数

  • 如果存在 ,使得当 时,有
那么级数绝对收敛
  • 如果对充分大的 ,有
那么级数发散。[1]

极限形式

对任意级数 ,令

  • 时级数绝对收敛
  • 时说明级数 发散(没有绝对收敛),原级数 可能收敛也可能发散。
  • 时级数可能收敛也可能发散[2][3]

证明

  • 时,存在 使得 . 则:
对充分大的

因为当 时级数 收敛,故级数 时收敛,即级数 绝对收敛。 [4]

  • 时,有
,则
,即
由于 发散,故 发散。[1]

例子

时无法判断其敛散性,举例如下:

已知有
已知当 时, ;当 时, ,然而由上式得
这说明当 时,拉比判别法无效。[5]

参考文献

  1. ^ 1.0 1.1 1.2 常庚哲,史济怀. 数学分析教程(下册). 安徽合肥: 中国科学技术大学出版社. 2013: 第173页. ISBN 9787312031311. 
  2. ^ 谢惠民. 数学分析习题课讲义. 北京: 高等教育出版社. 2004: 第8页. ISBN 9787040129410. 
  3. ^ Weisstein, Eric W. (编). Raabe's Test. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-09-02]. (原始内容存档于2015-04-02) (英语). 
  4. ^ Mathumatiks :: Raabes Test and Logarithmic Test. mathumatiks.org. [2015-09-03]. (原始内容存档于2016-03-04). 
  5. ^ Weisstein, Eric W. (编). Wolfram MathWorld (首頁). at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2015-09-02]. (原始内容存档于2015-09-05) (英语). 
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拉比判别法
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