A sequence is some collection of numbers (or more generally, objects) in a specific order. This is similar to Sets, however order matters and repetition is allowed. Formally a sequence is a function \(a: \mathbb{N} \mapsto X\), where \(X\) are the terms of the sequence. This is denoted \((a_n)_{n \in \mathbb{N}}\).
There does not need to be an obvious rule relating the terms. However, when there is it may be explicitly defined as \(a_{n}=f(n)\), for example the odd numbers \(a_{n}=2n+1 \to 1,3,4,5,9\ldots\). Or, it may be recursively defined as Recurrence relations.
A recursive sequence where each term is some common difference from the previous is an arithmetic progression. These take the form \[ a_{n} = d + a_{n-1} \] Any term may be found via \(a_{n} = a_{0} + n \cdot d\).
Where subsequent terms are related by a common ratio, the sequence is a geometric progression. \[ a_{n} = r \cdot a_{n-1} \] Terms may then be found directly as \(a_{n} = a_{0} \cdot r^n\).
A sequence is termed to be convergent when increasing \(n\) it approaches a finite constant value \[ \lim_{n\to \infty} a_{n} = L < \infty \] As an example, any geometric sequence where \(q<1\) will converge to \(0\).
If a sequence instead never reaches a finite value (either goes to \(\infty\) or oscillates) it is divergent.
Where the terms of a sequence are summed, the result is termed a series.
For example given the infinite sequence \((a_{n})\) consider the partial sum of its terms (\(a_{0},\ a_{0}+a_{1},\ a_{0}+a_{1}+a_{2},\ \ldots\)). This is expressed as \(\sum_{i=0}^{\infty} a_{i}\) which denotes both the series, and if the series is convergent the result of this summation.