# What does it mean for a sequence to converge?

A sequence that converges is one that adds to a number.

#### Explanation:

An infinite sequence of numbers can do 1 of 2 things - either converge or diverge, that is, either be added up to a single number (converge) or add up to infinity.

A series such as:

$1 + 2 + 3 + 4 + \ldots$ will diverge as adding it up will sum to $\infty$

as will

$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots$

But $\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots = \ln \left(2\right)$ and therefore is convergent

May 22, 2016

A sequence is said to converge if there is some particular number to which it tends.

#### Explanation:

Consider a sequence of Real numbers:

${a}_{1} , {a}_{2} , {a}_{3} , \ldots$

Such a sequence is said to converge to a limit $a$ if the difference between ${a}_{n}$ and $a$ eventually decreases to $0$ as $n$ increases.

Expressed in formal symbols:

$\forall \epsilon > 0 \exists N \in \mathbb{Z} : \forall n \ge N , \left\mid {a}_{n} - a \right\mid < \epsilon$

$\forall$ means "for all"

$\exists$ means "there exists"

With some words we could say:

If $\epsilon$ is any positive number (however small), then there is some integer $N$ such that the $N$th term of the sequence onwards are closer than $\epsilon$ to the value $a$.

If there is no number $a$ to which a sequence converges, then the sequence is said to diverge.

Examples

The sequence:

$1 , \frac{1}{2} , \frac{1}{3} , \frac{1}{4} , \ldots$ is convergent with limit $0$

The sequence (of Fibonacci ratios):

$\frac{1}{1} , \frac{2}{1} , \frac{3}{2} , \frac{5}{3} , \frac{8}{5} , \frac{13}{8} , \ldots$ is convergent with limit $\frac{1 + \sqrt{5}}{2} \approx 1.618034$

The sequence:

$1 , 2 , 3 , 4 , \ldots$ is divergent and unbounded.

The sequence:

$1 , - 1 , 1 , - 1 , 1 , - 1 , \ldots$ is divergent but bounded.