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狄利克雷判别法

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}

狄利克雷判别法(Dirichlet test)是一个级数审敛法,以数学家约翰·彼得·狄利克雷命名。

给定两个实数级数,如果级数满足

  • 对于所有正整数

其中M是某个常数,那么级数收敛。

狄利克雷判别法的一个推论,是更加常用的交错级数判别法

另外一个推论是当是一个趋于零的递减数列时, 收敛。

参考文献

  • Hardy, G. H., A Course of Pure Mathematics, Ninth edition, Cambridge University Press, 1946. (pp. 379-380).
  • Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis, Marcel Dekker, Inc., New York, 1981. (§8.B.13-15) ISBN 0-8247-6949-X.

外部链接

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狄利克雷判别法
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