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ContinuedFractionK[f,g,{i,imin,imax}]

表示连分数 .

ContinuedFractionK[g,{i,imin,imax}]

表示连分数 .

更多信息和选项
Details and Options Details and Options
范例  
基本范例  
选项  
GenerateConditions  
属性和关系  
可能存在的问题  
巧妙范例  
参见
相关指南
历史
按以下格式引用：

ContinuedFractionK

ContinuedFractionK[f,g,{i,imin,imax}]

表示连分数 .

ContinuedFractionK[g,{i,imin,imax}]

表示连分数 .

更多信息和选项

范例

打开所有单元 关闭所有单元

基本范例  (2)

一个简单的连分数：

Wolfram Language code: ContinuedFractionK[1, {n, 1, ∞}]

连分数的 ^(th) 收敛：

Wolfram Language code: ContinuedFractionK[n ^ 2, {n, 1, 10}]
Wolfram Language code: ContinuedFractionK[n ^ 2, {n, 1, 20}]

选项  (1)

GenerateConditions  (1)

生成连分数收敛的条件：

Wolfram Language code: ContinuedFractionK[a , {n, 1, ∞}, GenerateConditions -> True]

属性和关系  (2)

连分数可以用解与二阶递归方程的比值来得到：

Wolfram Language code: rec = v[n + 1] == b[n] v[n] + a[n] v[n - 1];
Wolfram Language code: u[m_, {v0_, v1_}] := First[RecurrenceTable[rec && v[0] == v0 && v[1] == v1, v, {n, m + 1, m + 1}]]

连分数是两个线性无关的解的比值：

Wolfram Language code: Table[(u[m, {1, 0}]/u[m, {0, 1}]) - ContinuedFractionK[a[n], b[n], {n, m}], {m, 6}]

ContinuedFractionK 和 FromContinuedFraction 互为倒数：

Wolfram Language code: Table[ContinuedFractionK[k ^ 6, {k, 1, n}], {n, 6}]
Wolfram Language code: Table[FromContinuedFraction[Table[k ^ 6, {k, 1, n}]], {n, 6}]
Wolfram Language code: % == 1 / %%

可能存在的问题  (1)

连分数可能不收敛：

Wolfram Language code: ContinuedFractionK[1 / n ^ 2, {n, 1, ∞}]

巧妙范例  (1)

创建一组连分数：

Wolfram Language code: flist = {{Cos[n], Sin[n], {n, 3}}, {1, 1 + n, {n, 1, a}}, {1, 1, {n, 1, ∞}}, {n, n, {n, 1, ∞}}, {n^2, 1 + 2 n, {n, 1, ∞}}, {1 + Mod[n, 2], {n, 1, ∞}}};
Wolfram Language code: Grid[Join[{{Text["Continued Fraction"], Text["Product"]}}, Transpose[{Inactive@ContinuedFractionK@@@flist, Map[ContinuedFractionK@@#&, flist]}]], IconizedObject[«Grid options»]]//TraditionalForm
Wolfram Research (2008)，ContinuedFractionK，Wolfram 语言函数，https://reference.wolfram.com/language/ref/ContinuedFractionK.html.

文本

Wolfram Research (2008)，ContinuedFractionK，Wolfram 语言函数，https://reference.wolfram.com/language/ref/ContinuedFractionK.html.

CMS

Wolfram 语言. 2008. "ContinuedFractionK." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/ContinuedFractionK.html.

APA

Wolfram 语言. (2008). ContinuedFractionK. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/ContinuedFractionK.html 年

BibTeX

@misc{reference.wolfram_2026_continuedfractionk, author="Wolfram Research", title="{ContinuedFractionK}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/ContinuedFractionK.html}", note=[Accessed: 09-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_continuedfractionk, organization={Wolfram Research}, title={ContinuedFractionK}, year={2008}, url={https://reference.wolfram.com/language/ref/ContinuedFractionK.html}, note=[Accessed: 09-September-2026]}