“geometric series”的概念、定义、习题解题方法及翻译-数学辞典

单词

geometric series

释义

geometric series

A real or complex, finite or infinite series a1 + a2 + a3 + …in which the terms form a geometric sequence. Thus the terms have a common ratio r with ak/ak−1 = r for all k. If the first term a1 equals a, then ak = ark−1. Let sn be the sum of the first n terms, so that sn = a + ar + ar2 + ⋯ + arn−1. Then sn is given (when r ≠ 1) by the formulae

If the common ratio r satisfies |r|<1, then rⁿ → 0 and it can be seen that sn → a/(1−r). The value a/(1−r) is called the sum to infinity of the series a + ar + ar2 + …. In particular, for |x|<1, the geometric series 1 + x + x2 + …has sum to infinity equal to 1/(1−x). For example, putting $inline$ , the series $inline$ as sum 2. If |x| ≥ 1, then sn does not tend to a limit and the series has no sum to infinity.

随便看

数学辞典收录了4151条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。