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
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.
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.
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( )
.
|
