Sequence

A sequence is basically a general list type mathematical tool that can be indexed. It's technically a function where the domain is a consecutive set of integers.

g: {1,2,3,4} -> R

g(1) = 3.98, g(2) = 4.0 etc...

Sequences can have positive or negative indexes.

They can be infinite or finite

For a sequence to be considered increasing or decreasing, every indices must be larger or smaller than the previous one, respectfully.

Formula

Sequences can be defined as a formula:

d(k) = 2^k where k = 0,1,2,3...

Geometric sequence

A sequence of real numbers where each element is obtained by multiplying the previous element by some common ratio.

a, ar, ar^2, ar^3, ar^4,...,ar^n

Compound interest is a geometric sequence

base_amount(1+rate)^years

Arithmetic sequence

A sequence of real numbers where each element is obtained by adding a constant to the previous element.

a, a+d, a+2d,..., a+nd

Summation notation

t (upper limit)
∑ a[i]              = a[s] + a[s + 1] + a[s + 2] + ... + a[t]
i = s (lower limit)

Sums can also sometimes be expressed without summation notation, this is called closed form:

Sum of terms in a geometric sequence for any real number r != 1 and any integer
n >= 1:

n-1
∑ (a*r^k) = [a(r^n - 1)]/(r - 1)
k=0

Recurrence relation

When you cannot create a general formula for a sequence and need to refer to the previous terms of the sequence

example: fibonacci sequence

f[0] = 0
f[1] = 1
f[n] = f[n-1] + f[n-2] for n >= 2

This is an example of a dynamical system where state changes over time and a discrete time dynamical system where time is divided into intervals and the state of the system is fixed during each interval and is a function of the previous time interval.

Limits

The limit of an infinite sequence is just where the sequence approaches as you use sequence elements close to the limit. Just think of it as a function of n. If you do not see a clear single grouping, the limit is not defined.