Question

What does it mean for a sequence to converge?

Answer 1

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+...` will diverge as adding it up will sum to `oo`

as will

`1/1+1/2+1/3+1/4+...`

But `1/1-1/2+1/3-1/4+...= ln(2)` and therefore is convergent

Answer 2

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 `a` if the difference between `a_n` and `a` eventually decreases to `0` as `n` increases.

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 `a` to which a sequence converges, then the sequence is said to diverge.

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.