裂项和

裂项求和（Telescoping sum）是一个非正式的用语，指一种用来计算级数的技巧：每项可以分拆，令上一项和下一项的某部分互相抵消，剩下头尾的项需要计算，从而求得级数和。

\sum_{i=1}^n(a_i-a_{i+1}) = (a_1-a_2)+(a_2-a_3)+\ldots+(a_n-a_{n+1})=a_1-a_{n+1}.

裂项积（Telescoping product）也是差不多的概念：

\prod_{i=1}^n \frac{a_i}{a_{i+1}} = \frac{a_1}{a_{n+1}}

单词	Telescoping series
释义	Telescoping series 中文百科裂项和裂项求和（Telescoping sum）是一个非正式的用语，指一种用来计算级数的技巧：每项可以分拆，令上一项和下一项的某部分互相抵消，剩下头尾的项需要计算，从而求得级数和。 $\sum_{i=1}^n(a_i-a_{i+1}) = (a_1-a_2)+(a_2-a_3)+\ldots+(a_n-a_{n+1})=a_1-a_{n+1}.$ 裂项积（Telescoping product）也是差不多的概念： $\prod_{i=1}^n \frac{a_i}{a_{i+1}} = \frac{a_1}{a_{n+1}}$ 英语百科 Telescoping series 裂项和 In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.
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Telescoping series