arithmetic series

Japanese: arithmetic series（英語表記）arithmeticseries

...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 [Arithmetic progression]

...The _sum Sn of the first to nth terms of this arithmetic progression is given by. The formula a1 + a2 ₊ ... + an ₊ ... (2) which formally joins the terms of an arithmetic progression _a1 , _a2 , ... , an , ... with an addition sign ( ₊ ) is called an arithmetic series, but since the sum of the infinite terms diverges except in the trivial case where _the first term and the common difference are both 0 (i.e., all a _n are 0), (2) has no concrete meaning. In general, the sum Sn of the first to nth terms of a series in the form of (2) is called the partial sum of the series, _and the formula for the partial sum of an arithmetic progression is none other than (1). ...

*Some of the terminology that refers to "arithmetic 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。…

【等差数列】より

…また，この等差数列の初項から第n項までの和S_nは，で与えられる。等差数列a₁,a₂，……，a_n，……の項の順に形式的に加号(＋)で結んだ式，　a₁＋a₂＋……＋a_n＋……　　……(2)　を等差級数arithmetic seriesというが，この無限個の項の和は初項も公差も0(すなわち，すべてのa_nが0)というつまらない場合を除き発散するから，(2)は具体的な意味をもたない。一般に(2)の形の級数の第1項から第n項までの和S_nを級数の部分和というが，等差数列の部分和の公式は(1)にほかならない。…