What is a Series?

A series is the sum of the terms of a sequence.

If the sequence is finite then the series is finite. If the sequence is infinite then the series is infinite.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. If the sequence has the form:

a_1, a_2, a_3, a_4, \ldots

then the series of the sequence is:

S = a_1 + a_2 + a_3 + a_4 + \ldots = \sum_{n=1}^{\infty} a_n

The sum of the first $n$ terms of a sequence is called the $n$ th partial sum. The $n$ th partial sum of a sequence is denoted by $s_n$ .

s_n = a_1 + a_2 + a_3 + \ldots + a_n = \sum_{i=1}^{n} a_i