Let's Recap

Chapter 2 So Far

• Sets
• Set operations
• Functions

Sequence

A function from a set of consecutive integers (typically the natural numbers or positive integers) to a set S. We use the notation an  to denote the image of the integer n. an  is also the nth term of the sequence.

Can be finite or infinite.

Sequence

A set of integers typically generated by a function. Note that sets are NOT ordered, but we typically talk about sequences being a function of an index n.

Examples of Sequences

• a_n = 1 / n
• a_n = b * rⁿ (geometric progression) 
• a_n = b + (n * d) (arithmetic progression) 

Recurrence Relations

Recurrence Relations

A sequence defined in terms of the sequence's previous values. We can also say that a recurrence relation is recursively defined.

Recurrence relations typically have a boundary condition which establishes the initial terms of the sequence.

Examples of Recurrences

• f_n = f_n-1 + n where n > 0
• f_n = fn - 1 + fn-2 where f1 = 1 and f2 = 1

Solutions of a Recurrence

A sequence is called a solution if the terms of the sequence satisfy the recurrence relation.

Example: {1, 1, 2, 3, 5, 8, 13, ...}

A closed formula is a solution to the relation expressed as a non-recursive function of n.

Example: f(n) = n * (n + 1) / 2

Summations / Series

Series

A sum of all of the terms in a sequence. Can be a partial sum (a sum of the first n terms). Series can also be finite or infinite.

Σⁿ_i=m s_i denotes the sum from m to n.

Closed Formula

Just like sequences, series can have a closed formula where a solution is represented only in terms of its bounds.

In mathematical terms,

f(n) = Σⁿ_i=m s_i 

Examples of Series

• Σⁿ_i=0 i = n * (n + 1) / 2 
• Σ^∞_i=0 xⁱ = 1/(1 - x) where | x | < 1
• Σ^∞_k=1 1/k (The harmonic series)
• Σ⁵_i=1Σ⁵_j=1 i*j