# What does it mean for a sequence to converge?

##### 2 Answers

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:

as will

But

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,...#

Such a sequence is said to converge to a limit

Expressed in formal symbols:

#AA epsilon > 0 EE N in ZZ : AA n >= N, abs(a_n - a) < epsilon#

#AA# means "for all"

#EE# 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

**Examples**

The sequence:

#1, 1/2, 1/3, 1/4,...# is convergent with limit#0#

The sequence (of Fibonacci ratios):

#1/1, 2/1, 3/2, 5/3, 8/5, 13/8,...# is convergent with limit#(1+sqrt(5))/2 ~~ 1.618034#

The sequence:

#1, 2, 3, 4,...# is divergent and unbounded.

The sequence:

#1, -1, 1, -1, 1, -1,...# is divergent but bounded.