geometric series

Japanese: geometric series（英語表記）geometricseries

...Each element that makes up a sequence of numbers, functions, or series is called a term of that sequence, function, or series. A series in which the difference between each term and the next term is constant, as in the first example above, is called an arithmetic series, and a series in which the ratio between each term and the next term is constant, as in the second example, is called a geometric series. Also, a series whose terms are the reciprocals of each term in an arithmetic series is called a harmonic series. ...

From [Geometric sequence]

…The _sum Sn of the first to nth terms of this geometric sequence is given by. The equation a + ar + ar ² + ... + ar ^{n -1} + ... (2) which formally joins the geometric sequence a , ar , ar ² , ..., ar ^{n -1} , ... in this order with addition signs (+) is called a geometric series, but when | r | ≧ 1, this infinite series does not converge, so (2) has no concrete meaning. If | r | < 1, r ⁿ approaches 0 infinitely as n becomes infinitely large, so the value _of Sn in (1) approaches S = a /(1 - r ). …

*Some of the terminology that refers to "geometric series" is listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…数列，関数列または級数を構成する各要素を，その数列，関数列または級数の項という。上の第1の例のように各項とその次の項との差が一定である級数を等差級数arithmetic seriesまたは算術級数といい，第2の例のように各項とその次の項との比が一定である級数を等比級数geometric seriesまたは幾何級数という。また，等差級数の各項の逆数を項とする級数を調和級数harmonic seriesという。…

【等比数列】より

…また，この等比数列の初項から第n項までの和S_nは，で与えられる。等比数列a,ar,ar²，……，ar^n－1，……をこの順に形式的に加号(＋)で結んだ式，　a＋ar＋ar²＋……＋ar^n－1＋……　　……(2)　を等比級数geometric seriesというが，｜r｜≧1のときはこの無限級数は収束しないから，(2)は具体的な意味をもたない。｜r｜＜1ならばnが限りなく大きくなるときrⁿは0に限りなく近づくから，(1)のS_nの値は，S＝a/(1－r)に限りなく近づく。…