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} = 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.
Limit of an Infinite Geometric Series

For a geometric sequence a_{n} = a_{1}r^{n1}, where 1 < r < 1, the limit
of the infinite geometric series a_{1}r^{n1} = . This is the same as the sum of the infinite geometric
sequence a_{n} = a_{1}r^{n1}.

Sum of a Finite Arithmetic Sequence

The sum of the first n terms of the arithmetic sequence is S_{n} = n() or S_{n} = na_{1} + (dn  d ), where d is the
difference between each term.

Sum of a Finite Geometric Sequence

For a geometric sequence a_{n} = a_{1}r^{n1}, the sum of the first n
terms is S_{n} = a_{1}().
