In mathematics, a sequence can be thought of as a list of numbers written in a definite order.

$a_{1}, a_{2}, a_{3}, a_{4}, ..., a_{n}, ...$

Note that the sequence ${a_{1}, a_{2}, a_{3}, a_{4},}$ ... can also be written as ${a_{n}}_{n = 1}^{\infty}$ or ${a_{n} : n = 1, 2, 3, ...}$ .
Some sequences do not have a simple defining analytical expression and some sequences can be defined simply by giving a formula for the nth term. For example, from the first few terms of the sequence $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5},$ ... , we observe that, if this pattern continues, then the general term can be written as $a_{n} = \frac{n}{n + 1}$ .

Definition: An infinite real series is an expression of the form $\sum_{n = 1}^{\infty} a_{n} = a_{1} + a_{2} + a_{3} + ... + a_{i} + ...$
where $a_{n}$ is an infinite sequence of real numbers. For example $\sum_{n = 1}^{\infty} = \frac{1}{n^{2}} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + ... + \frac{1}{i^{2}} + ...$

Definition: The $n^{t} h$ partial sum $S_{n}$ of the series is the sum of its first $n$ terms. $(S_{n} = a_{1} + a_{2} + a_{3} + ... + a_{n} .)$

This in turn defines the sequence $S_{1}, S_{2}, S_{3}, S_{4}, ..., S_{n}$ which is called the sequence of partial sums of the series $\sum_{n = 1}^{\infty} a_{n}$ .

Definition: We say that the infinite series $\sum_{n = 1}^{\infty} a_{n}$ . converges (or is convergent) with sum S provided that the limit of its sequence of partial sums, $lim_{n \to \infty} S_{n} = S$ exists and is finite. Otherwise we say that the series diverges or is divergent. Note that if a series is divergent, it does not have a finite sum because $S_{n}$ does not converge to any finite number $S$ .