Sequences and Series: Terms and Formulae

Terms

Arithmetic Sequence

A sequence in which each term is a constant amount greater or less than the previous term. In this type of sequence, a_n+1 = a_n + d, where d is a constant.
Common Ratio

In a geometric sequence, the ratio r between each term and the previous term.
Convergent Series

A series whose limit as n→∞ is a real number.
Divergent Series

A series whose limit as n→∞ is either ∞ or - ∞.
Explicit Formula

A formula for the nth term of a sequence of the form a_n = some function of n.
Finite Sequence

A sequence which is defined only for positive integers less than or equal to a certain given integer.
Finite Series

A series which is defined only for positive integers less than or equal to a certain given integer.
Geometric Sequence

A sequence in which the ratio between each term and the previous term is a constant ratio.
Index of Summation

The variable in the subscript of Σ. For a_n, i is the index of summation.
Infinite Sequence

A sequence which is defined for all positive integers.
Infinite Series

A series which is defined for all positive integers.
Recursive Sequence

A sequence in which a general term is defined as a function of one or more of the preceding terms. A sequence is typically defined recursively by giving the first term, and the formula for any term a_n+1 after the first term.
Sequence

A function which is defined for the positive integers.
Series

A sequence in which the terms are summed, not just listed.
Summation Notation

a_n = a₁ + a₂ + a₃ + a₄ + ... + a_n. The symbol Σ and its subscript and superscript are the components of summation notation.
Term

An element in the range of a sequence. A sequence is rarely represented by ordered pairs, but instead by a list of its terms.