查看“︁無窮乘積”︁的源代码

在[[數學]]中，對於複數[[序列]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ...，'''無窮乘積''' 
:<math>
\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots
</math>

定義為部分乘積''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub>在''n''的增加沒有邊界時的[[序列的極限|極限]]。當這個極限存在並且不是0的時候，這個乘積稱為“收斂”，否則稱為發散。

== 收斂條件 ==
正實數的乘積 <math>\prod_{n=1}^{\infty} a_n</math> 收斂，若且唯若 <math>\sum_{n=1}^{\infty} \log a_n</math> 收斂。

==參見==
*[[無窮級數]]
*[[連分數]]
*[[魏尔施特拉斯分解定理]]

== 參考 ==
* {{cite book
|last=Knopp
|first=Konrad
|authorlink=Konrad Knopp
|title=Theory and Application of Infinite Series
|url=https://archive.org/details/theoryapplicatio0000knop_w8t7
|publisher=[[Dover Publications]]
|year=1990
|isbn=978-0-486-66165-0
|language=en
}}
* {{cite book
|last=Rudin
|first=Walter
|authorlink=Walter Rudin
|title=Real and Complex Analysis
|url=https://archive.org/details/realcomplexanaly0000rudi
|edition=3rd
|publisher=[[McGraw Hill]]
|location=Boston
|year=1987
|isbn=0-07-054234-1
}}
* {{Cite book
|editor1-last=Abramowitz
|editor1-first=Milton
|editor1-link=Milton Abramowitz
|editor2-last=Stegun
|editor2-first=Irene A.
|editor2-link=Irene Stegun
|title=[[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]
|publisher=[[Dover Publications]]
|year=1972
|isbn=978-0-486-61272-0
}}

==外部連結==
*[http://mathworld.wolfram.com/InfiniteProduct.html Infinite products from Wolfram Math World] {{Wayback|url=http://mathworld.wolfram.com/InfiniteProduct.html |date=20210416070626 }}

[[Category:序列]]


{{數學小作品}}