home

# Sequences and series

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.

## Arithmetic sequence

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 $$a_{n} = a_{0} + n \cdot d$$.

## Geometric sequence

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 $$a_{n} = a_{0} \cdot r^n$$.

## Limits

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.

## Series

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.