# Sequences & Series

Terms
Sequence
a list of numbers, usually separated by commas (t1, t2, t3, ...) (Technically, a sequence is a function whose domain is the natural numbers.)
Term
one of the numbers in a sequence or series
Arithmetic Sequence
a sequence in which the difference between successive terms is constant
Common Difference
the difference between successive terms in an arithmetic sequence

Geometric Sequence

a sequence in which the ratio of successive terms is constant
Common Ratio
the ratio of successive terms in a geometric sequence
Finite Sequence
a sequence that has a last term
Infinite Sequence
a sequence that does not have a last term
Series
the sum of a sequence (t1 + t2 + t3 + ...)
Arithmetic Series
a series based on an arithmetic sequence (the sum of the terms of an arithmetic sequence)
Geometric Series
a series based on a geometric sequence (the sum of the terms of a geometric sequence)
Finite Series
a series based on a finite sequence (the sum of the terms of a finite sequence)
Infinite Series
a series based on an infinite sequence (the sum of the terms of an infinite sequence)
Partial Sum
the sum of a finite number of terms of a series
Sequence of Partial Sums
every infinite series has a sequence of partial sums = first term, sum of first two terms, sum of first three terms, sum of first four terms, etc.

Converges

If the sum of an infinite series has a limit, the series converges.
Diverges
If the sum of an infinite series has no limit, the series diverges.
Explicit Definition
tells how to calculate any specified term in a sequence or series
Implicit Definition
tells how to construct a sequence or series by building it term-by-term starting with the first term
Limit
a number that the terms of a sequence (or partial sums of a series) gets closer to as the number of terms increases
n approaches infinity
the number n gets larger and larger (forever)
limit of some expression equals infinity the value of the expression gets larger and larger (forever)

Symbols
n the number of the current (or next) term in a sequence or series
t the value of a term in a sequence or series
d the common difference of an arithmetic sequence or series
r the common ratio of an geometic sequence or series
tn the value of the current (or next) term in a sequence or series
tn-1 the value of the previous term in a sequence or series (the term before the nth term)
S the value of a series (the sum of terms)
Sn the (partial) sum of the first n terms of a series
Sn-1 the (partial) sum of the first n - 1 terms of a series (the sum of all the terms before term number n) What happens to the value of the expression as the number of terms in its sequence (or series) gets larger and larger?

Recursive Definitions
Step 1 Where does the sequence or series start? What is the first term (t1)?
Step 2

How do you calculate the next term (tn ) of the sequence or series based on the terms you have so far?

The last term so far is tn-1.

The second to the last term so far is tn-2, etc.

Formulas
Explicit Definition of an Arithmetic Sequence
tn = t1 + (n - 1)d
Explicit Definition of a Geometric Sequence
tn = t1rn-1
Sum of a Finite Arithmetic Series Sum of a Finite Geometric Series Limit of a Rational Function Limit of a Power Function Sum of an Infinite Geometric Series last update December 18, 2007 by JL Stanbrough