Sequences and Series

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 n th 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 = a ₁ + a ₂ + a ₃ + a ₄ + ... + 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.