Sequences |

**In General**

$ __Def__: A sequence in a set *X*
is a subset *A* ⊂ *X* and a map \(\mathbb N\) → *A*,
*n* \(\mapsto\) *a _{n}*, which is
onto, so that

>

**Operations on Sequence**s

$ __Difference operator__:
The mapping {*a*_{n}}
→ {*b*_{n}}
defined by *b*_{n}
= *a*_{n+1} −
*a*_{n} for all *n*.

**Types of Sequences**

$ __Cauchy sequence__:
A sequence {*a*_{n}}
in a normed space such that, for all *ε* > 0, ∃
*N* such that for all *n*, *m* > *N*,
|| *a*_{n}−
*a*_{m}|| < *ε*.

$ __Monotonic sequence__: (Increasing)
One such that the difference operator produces a non-negative sequence.

$ __Completely monotonic sequence__:
(Increasing) One such that *k* iterations of the difference operator produce
a non-negative sequence, for all *k*.

**Convergence** > s.a. limit;
series.

$ __In a topological space ( X,
T)__: A sequence {

$

$

$

**Fibonacci Sequence** > s.a. Golden Ratio.

$ __Def__:
The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,
..., in which *F*_{n}
= *F*_{n−2}
+ *F*_{n−1};
It appears in life patterns.

* __Result__: (Zeckendorf) Every
integer can be written uniquely as a sum of non-consecutive Fibonacci numbers.

@ __General references__:
Vorobiev 02 [IIb];
Posamentier & Lehmann 07;
Miller & Wang JCTA(12) [and central-limit-type theorems].

@ __In nature__:
news pw(07)apr [stress-related Fibonacci spirals];
Pennybacker & Newell PRL(13) [underlying physical and biochemical mechanisms in plant growth];
Faraoni & Atieh a2101
[continuous generalizations and cosmological analogies].

> __Online resources__:
See Wikipedia page.

**Other Examples**
> see Wikipedia page on integer sequences.

**Generalizations** > s.a. projective family.

* __Inductive family or system__:
Given a directed set *I*, an inductive family on *I* is
a collection {*X*_{i}
| *i* ∈ *I*} of objects in a category, and immersions
{*φ*_{ij}:
*X*_{i} →
*X*_{i} |
*i* ≥ *j* ∈ *I*}, such that
*φ*_{ii}
= id_{Xi}, and
*φ*_{ij} \(\circ\)
*φ*_{jk} =
*φ*_{ik}; __Remark__:
In many (all?) categories each such family defines an inductive limit.

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send feedback and suggestions to bombelli at olemiss.edu – modified 28 jan 2021