Discrete Math Chapter 8: Sequences and Summations.

Expanded Form (of the sum)

1^3 + 2^3 + 3^3 + 4^3.

Closed Form (of the sum)

A mathematical expression that expresses the value of the sum without summation notation. There are closed form expressions for some, although not all, of the summations that arise naturally in scientific applications.

Explicit Formula

A sequence can be specified by an explicit formula showing how the value of term a(k) depends on k. For example, d(k) = 2^k for k ≥ 1. The infinite sequence {d(k)} starts with: 2, 4, 8, 16, ...

Sequence

A sequence is a special type of function in which the domain is a consecutive set of integers. For example, a sequence can be defined to denote a student's GPA for each of the four years the student attended college. The domain of the function is {1, 2, 3, 4} for each of the four years.

Arithmetic Sequence

A sequence of real numbers where each term after the initial term is found by taking the previous term and ADDING a fixed number called the common difference. An arithmetic sequence can be finite or infinite.

Geometric Sequence

A sequence of real numbers where each term after the initial term is found by taking the previous term and MULTIPLYING by a fixed number called the common ratio. A geometric sequence can be finite or infinite.

Finite Sequence

A sequence with a finite domain.

Infinite Sequence

A sequence with an infinite domain.

{g(k)}

Entire sequence of g(k).

Term

Ex: g(x) (when x is a subscript), g(x) is the term.

a(n)

Final term.

a(m)

Initial term.

g(k) (without curly braces)

Refers to a single term in the sequence.

Summation Notation

Used to express the sum of terms in a numerical sequence.

Index

Variable of a sequence. Ex: g(x) (when x is a subscript), x is the index.

Increasing Sequence

Whenever the sequence is defined on two consecutive indices k and k+1, then a(k) < a(k+1).

Non-Decreasing Sequence

Whenever the sequence is defined on two consecutive indices k and k+1, then a(k) ≤ a(k+1).

Initial Index

m.

Final Index

n, where n >=m.

Index of summation

t∑i=st a(i)=a(s)+a(s+1)+⋯+a(t): The variable i is called the index of summation. Any variable name could be used instead of i, but variables i, j, k, and l are the most common.

Lower Limit

t∑i=st a(i)=a(s)+a(s+1)+⋯+a(t): The variable s is the lower limit of the summation.

Upper Limit

t∑i=st a(i)=a(s)+a(s+1)+⋯+a(t): The variable t is the upper limit of the summation.

Summation Form (of the sum)

∑4j=1 j^3.