Sequences and series

A sequence is a list of terms, while a series is the sum of the terms. A sequence can be defined in two ways:

A recurrence relation, e.g. (for the Fibonacci numbers)
A position-to-term formula, e.g. (for an arithmetic sequence). A sequence with a position-to-term formula can be denoted as , e.g. .

Sequences can be:

Periodic, meaning that there are repeated terms, e.g. . In this example, the period is 3.
Oscillating, meaning they are periodic with just two terms, e.g. .
Convergent, meaning the terms of the sequence get closer to a limiting value, e.g. the terms in converge to 0. A series can also be convergent, meaning the sum to infinity of the sequence has a finite value^[1]. For example, the sum of converges to 2.
Divergent, meaning the terms of the sequence are not convergent, e.g.
A series can also be divergent, meaning the sum to infinity of the sequence does not have a finite value.
Monotonically increasing, meaning each term is greater than the one before it (), e.g. . A monotonically decreasing sequence has each term less than the one before it ().

The sequence of Fibonacci numbers is defined by the recurrence relation , , , giving the first 5 terms as . The ratio of each term to the previous one converges to , the golden ratio.

The sequence of Lucas numbers^[2] is defined by the same recurrence relation as the Fibonacci numbers (each term is the sum of the two immediately before it), but and , giving the first 5 terms as .

First-order recurrence relations

A linear recurrence relation is where each term in a sequence is a linear function of previous terms, e.g. . A first-order recurrence relation is where each term only depends on the previous term and some function of , e.g. . For the AS content, only first-order recurrence relations will be considered.

A homogenous recurrence relation has the form , for some constant . In contrast, a non-homogenous recurrence relation has the form , where is some function of (in this course, a polynomial in or ).

A recurrence system is defined by the recurrence relation (e.g. ) and the initial conditions, e.g. . Our goal is typically to find a closed-form solution to the system: a position-to-term rule that only depends on . For the previous example, (you can verify this for ).

For a relation of form , the solution will take the form: , where is the complementary function and is the particular solution.

We start by finding the complementary function , which is related to the homogenous part of the recurrence relation ():

Start with the reduced equation
Replace with (), giving the auxiliary equation
Solve for , giving a complementary function of the form
In the AS course, is always of the form . Note that if , this will always be a failure case.

We then find the particular solution , which is related to the non-homogenous part of the recurrence relation ():
Choose a form for based off :

If ) is a number,
If ) is linear,
If ) is quadratic,
If ) is an exponential (some constants and ),
If the form of is the same as for , then this is a failure case. Multiply by .
Substitute the appropriate form for into both sides of the relation and solve for , , and .

Then, apply the initial conditions to find remaining constants from the complementary function.

Footnote on convergence (outside of specification)
There are sequences where the terms converge to 0 that do not sum to a convergent series. For example, the harmonic sequence is convergent because the terms tend to 0. However, the harmonic series is not convergent.↩︎
Footnote on the Fibonacci and Lucas numbers
is more commonly used to denote the golden ratio than , but the specification uses .
Some definitions of the Fibonacci and Lucas numbers start from instead of (as an extra sidenote, Fibonacci himself started from 1 and 2).↩︎