# Sequences and Series

## Contents

#### 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 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.

#### Formulae

 Limit of an Infinite Geometric Series For a geometric sequence a n = a 1 r n-1 , where -1 < r < 1 , the limit of the infinite geometric series a 1 r n-1 = . This is the same as the sum of the infinite geometric sequence a n = a 1 r n-1 .

 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 n-1 , the sum of the first n terms is S n = a 1() .